attele THE UNIVERSITY OF ILLINOIS LIBRARY Sh eal THEMATICS LIBRA a ‘Return this book on or before the a Latest Date stamped below. A fod | _ charge is made on all overdue nay 26 wal. : ign i io. WORKS OF PROF. L. E. DICKSON PUBLISHED BY JOHN WILEY 3k SONGS: Introduction to the Theory of Algebraic Equa= tions. Small 8vo, v + 104 pages. Cloth, $1.25 net. College Algebra. A text-book for colleges and technical schools, Small 8vo, vii+ 214 pages. Illustrated. Cloth, $1.50 net. PUBLISHED BY B. G. TEUBNER, Leipzic, GERMANY. Linear Groups with an Exposition of the Galois Field Theory. 8vo, x + 312 pages. Cloth, 12 marks. INTRODUCTION TO THE es Oe YO) ALGEBRAIC EQUATIONS. BY LEONARD EUGENE DICKSON, Pu.D., ASSISTANT PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO, ETS PT ae LON, FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS. LonpoN: CHAPMAN & HALL, LIMITED. 1903. oe aes Copyright, 1903, BY L. E. DICKSON. - ROBERT DRUMMOND, PRINTER, NEW YORK. rey Yd ae PREFACE. THE solution of the general quadratic equation was known as early as the ninth century; that of the general cubic and quartic equations was discovered in the sixteenth century. During the suc- ceeding two centuries many unsuccessful attempts were made to solve the general equations of the fifth and higher degrees. In 1770 Lagrange analyzed the methods of his predecessors and traced all their results to one principle, that of rational resolvents, and proved that the general quintic equation cannot be solved by rational re- solvents. The impossibility of the algebraic solution of the general equation of degree n(n >4), whether by rational or irrational resoly- ents, was then proved by Abel, Wantzel, and Galois. Out of these algebraic investigations grew the theory of substitutions and groups. The first systematic study of substitutions was made by Cauchy (Journal de I’ école polytechnique, 1815). The subject is here presented in the historical order of its devel- opment. The First Part (pp. 1-41) is devoted to the Lagrange- Cauchy-Ahel theory of general algebraic equations. The Second Part (pp. 42-98) is devoted to Galois’ theory of algebraic equations, whether with arbitrary or special coefficients. The aim has been to make the presentation strictly elementary, with practically no dependence upon any branch of mathematics beyond elementary algebra. There occur numerous illustrative examples, as well as sets of elementary exercises. In the preparation of this book, the author has consulted, in addition to various articles in the journals, the following treatises: iii 685414 iv PREPAGE: Lagrange, Réflexions sur la résolution algébrique des équations; Jordan, T'raité des substitutions et des équations algébriques; Serret, Cours d’ Algéebre supérieure; Netto-Cole, T'heory of Substitutions and us Applications to Algebra; Weber, Lehrbuch der Algebra; Burn- side, The Theory of Groups Pierpont, Galois’ Theory of Algebrare Equations, Annals of Math., 2d ser., vols. 1 and 2; Bolza, On the Theory of Substitution-Groups and its Applications to Algebraic Equations, Amer. Journ. Math., vol. XIII. The author takes this opportunity to express his indebtedness to the following lecturers whose courses in group theory he has at- tended: Oscar Bolza in 1894, E. H. Moore in 1895, Sophus Lie in 1896, Camille Jordan in 1897. But, of all the sources, the lectures and publications of Professor Bolza have been of the greatest aid to the author. In particular, the examples (§ 65) of the group of an equation have been borrowed with his permission from his lectures. The present elementary presentation of the theory is the out- come of lectures delivered by the author in 1897 at the University of California, in 1899 at the University of Texas, and twice in 1902 at the University of Chicago. . CuicaGco, August, 1902 TABLE OF CONTENTS. CHAPTER PAGES I. Solution of the General Quadratic, Cubic, and Quartic Equa- tions. Lagrange’s Theorem on the Irrationalities Entering RN ye as aie ba es oie o/c ldg's aw a oe « coats MiNemelT «8 1-9 MENT 2 oe see Deiole sc o'h.<6 FG ee as 0.0 se aie oe ues 4 II. Substitutions; Rational Functions...... abet s MME es 5 . 10-14 | SME Feo cre he cacy iri aves aiding ce vila suites cngecae ss : 14 Ht) Substitution Groups; Rational Functions..........ccececcs . 15-26 Sau fre 5 LEP re Che 5 aio. in vee We 90 0/ne eels : 20 IV. The General Equation from the Group Standpoint........... 27-41 Ms fete ne Reig ce eae Os sess ont ghee 41 WV eeearenraic Introduction to Galois’Theory........sccecssesece 42-47 PeeeererroUp Ol an Piquation: ¢ ccc ccc cocewtescecesveseucs 48-63 rer ae Sa eed pies ce vd acess ch tesneevevens 57-58 VII. Solution by means of Resolvent Equations.............006. 64-72 VIII. Regular Cyclic Equations; Abelian Equations.............. 73-78 Seeeeriecrionstor Alecbraic Solvability..........ccccssccecvcces 79-86 X. Metacyclic Equations; Galoisian Equations......... miata «cette 87-93 XI. An Account of More Technical Results......... Rheate nee sles oe-OS APPENDIX. Pymamdecric Functions, .......2.esesee0e Mousseeocsescevec en Uo LUL On the General Equation ............. rive waves eecle seueew st Ulla INDEX... Tie caias cicidicisisietbia cons tcweeeeceeee coetveccuonlUs Vv THEORY OF ALGEBRAIC EQUATIONS. FIRST PART. THE LAGRANGE-ABEL-CAUCHY THEORY OF GENERAL ALGEBRAIC EQUATIONS. CHAPTER I. SOLUTION OF THE GENERAL QUADRATIC, CUBIC, AND QUARTIC EQUATIONS. LAGRANGE’S THEOREM* ON THE IRRATION- ALITIES ENTERING THE ROOTS. 1. Quadratic equation. The roots of x?+pr+q=0 are 2,=4(—ptV p?—4q), %,=4(—p—V p?—4q). By addition, subtraction, he multiplication, we get t+ 2, = Te Shes tama oe V p?—4q, Uyle= q. Hence the irrationality Vp?—4q, which occurs in the expressions for the roots, is rationally expressible in terms of the roots, being equal to x,—a,. Unlike the last function, the functions x,+2, and 2,7, are symmetric in the roots and are rational functions of the coefficients. 2. Cubic equation. The general cubic equation may be written (1) x? —c,x?+¢,%—¢c,=0. Setting c=y+4c,, the equation (1) takes the simpler form (2) y*+ pyt+q=9, * Reflexions sur la résolution algébrique des équations, Giuvres de Lagrange, Paris, 1869, vol. 3; first printed by the Berlin Academy, 1770-71. GENERAL QUADRATIC, CUBIC, AND QUARTIC. [Cu. I if we make use of the abbreviations (3) P=O—$C", Q=—C,+$C,0,—¥yC,*. The cubic (2), lacking the square of the unknown quantity, is called the reduced cubic equation. When it is solved, the roots of (1) are found by the relation x=y+ 4¢,. The cubic (2) was first solved by Scipio Ferreo before 1505. The solution was rediscovered by Tartaglia and imparted to * Cardan under promises of secrecy. But Cardan broke his promises and published the rules in 1545 in his Ars Magna, so that the formule bear the name of Cardan. The following method of deriving them is essentially that given by Hudde in 1650. By the transformation (4) y=2—, the cubic (2) becomes 23— Ae +q=0, whence ~ p? (5) 2+ gz? — 5 =0. Solving the latter as a quadratic equation for 23, we get 2=—hgtVR, R=1¢?+ayp. Denote a definite one of the cube roots of —4qg+V R by / —MtVR. The other two cube roots are then WY -WtVR, 0 ¥/—3g+VR, where w is an imaginary cube root of unity found as follows. The three cube roots of unity are the roots of the equation re—1=0, or (r—1)(r?+r+1)=0. The roots of r?+r+1=0 are —$+4V —3=o and —4—4V —3=@?, Then (6) w?+w+1=0, =k, s Src. 2] THEORY OF ALGEBRAIC EQUATIONS. 3 In view of the relation (—49+V R)(—3g-V R) =4@?— R= — dp’, a particular cube root Aye, R may be chosen so that aig VR. e/ q-V R= - 3 w 2/3 + VR - 0 &/'—4qg-V R= — 4p, w? Y/ At VR w8/—3q—-VR = — 4p. Hence the six roots of equation (5) may be separated into pairs in such a way that the product of two in any pair is —3$p. The root paired with z is therefore =2, and their sum z= is, in. view of (4), a root y of the cubic (2). In particular, the two roots of a pair lead to the same value of y, so that the siz roots of (5) lead to only three roots of the cubic, thereby explaining an apparent difficulty. Since the sum of the two roots of any pair of roots of (5) leads to a root of the cubic (2), we obtain Cardan’s formuls for the roots y,, Yy, yz of (2): y= —Ht+VR+4Y/ -hq-VF, (7) yaw 8/ —4g+V R+w? 8&/ —3q—V R, ts) — hgtVR+w 2/ — 4q—V R. Multiplying these expressions by 1, w?, w and adding, we get, by (6), ie Y -MAV RABY Foy toys)= Xo Using the multipliers 1, w, w?, we get, similarly, Y= hq-VR=4Y, + oy Foy). Cubing these two expressions and subtracting the results, we get VR= Fe { (ys + o7Y2+ wy)? — (Yr t oY, +0"ys)*} =¥—3y, wu wo Ys—Y3)) 4 GENERAL QUADRATIC, CUBIC, AND QUARTIC. [Cu I upon applying the Factor Theorem and the identity w—w?=V —3. Hence all the irrationalities occurring in the roots (7) are rationally expressible in terms of the roots, a result first shown by Lagrange. The function (Y1— Yo) (Y2— Ys) (Ys— V1)" = — 21 — 4p is called the discriminant of the cubic (2). The roots of the general cubic (1) are Lr=Yz+SCQy, Le=Yot Cy, Te=Y3+ Hey. ve Ly Le=Yr— Yo, Ly —XLE=Y2— Ya, %3—M%=Y3s—- 1; (8) (XL — X_)(L_—Xg)(X3—2%,) = (Y, —Y-)(Yo— Ys) Yea a) = VV R= 3 VP EXERCISES. 1. Show that 2, +077, + 02,=Yy,;+ 07y,+ wys, ©, + W2,+ W723 =Y; + OY, + W7Ys. ~ 2. The cubic (2) has one real root and two imaginary roots if R>0O; three real roots, two of which are equal, if R=0; three real and distinct roots if R<0O (the so-called irreducible case). 3. Show that the discriminant (x,—2,)?(x,—23)?(a3—2,)”. of the cubic (1) equals C,2c,? + 18¢,c.¢, —4c,3 —4c, °c, —27¢,7. Hint: Use formula (8) in connection with (3). 4. Show that the nine expressions 3/ =a Ee R +2/ Shoe R, where all combinations of the cube roots are taken, are the roots of the cubics yitpytq=0, y'twpyt+q=0, y*+w*pytq=0. ~5. Show that y,t+y.t+y3=0, YYotYYstyYs=P, YWYYs=—4- ‘6 Show that 7,142.42, =¢,, 0,%,+2,T%+2p%3=Cy, 1%o%, =, US How may these results be derived directly from equation (1)? 3. Aside from the factor 4, the roots of the sextic (5) are ~ $,=%,+wr,+0*Xs, Py=X,+ Wt, + way, | (r= =X, +0t,+W'X,, $s=0°~=T,+W%,+0"X,, $3= WP, =U + OX, +W°X,, Jo=Wby=X,+ Wk, + 07s. These functions differ only in the permutations of 2%,, 2, x. As there are just six permutations of three letters, these functions Src. 3] THEORY OF ALGEBRAIC EQUATIONS. 5 give all that can be obtained from ¢, by permuting 2,, 2,, x. For this reason, ¢, is called a six-valuwed function. Lagrange’s @ priori solution of the general cubic (1) consists in determining these six functions ¢,,...,¢, directly. They are the roots of the sextic equation (t—¢,)...(t—d¢,)=0, whose coefficients are symmetric functions of ¢,,..., ¢, and consequently symmetric functions of x,, 7,7, and hence * are rationally expressible in terms of ¢,, G, ¢,. Since ¢,=wd,, d,=wy,, etc., we have by (6) (t —,)(t —Po)(t +br?+cr+d=0, may be written in the form (a? + 40x)? = (4a?—b)a?—cx—d. With Ferrari, we add (x?+4ax)y+4y? to each member. Then (13) (x? +4ax-+4y)?=(40?—b+y)x? + (Fay—c)a+hy’—d. We seek a value y, of y such that the second member of (13) shall be a perfect square. Set (14) a’—4b+4y,=#?. The condition for a perfect square requires that als = 2 (5) H+ Gay, oe tay —d= (Je EZ)” ay tie zay,—C€\* __ (ay— 6)” Be t a?—4b+4y," Hence y, must be a root of the cubic, called the resolvent, (16) y®— by? + (ac—4d)y—a?d+ 4bd—c?=0. In view of (15), equation (13) leads to the two quadratic equations (17) x’? + (ga—3t)x + dy, — (Say, —c)/t=0, (18) w+ (Fat Ht)x + by, + (Gay, —©)/t=0. Let x, and x, be the roots of (17), x, and x, the roots of (18). Then LZ +X,= —Zatht, Xx,=sy,— (gay, —¢)/t, %g+2,=—ta—ht, 2x,0,=4y,+ (4ay,—0)/t. i Src. 5] THEORY OF ALGEBRAIC EQUATIONS. 7 By addition and subtraction, we get (19) y+ X,—XLz—Uy=b, My, + UzL~= Yj. In solving (17) and (18), two radicals are introduced, one equal to a,—2, and the other equal to z,—2, (see § 1). Hence all the irrationalities entering the expressions for the roots of the general quartic are rational functions of its roots. If, instead of y,, another root of the resolvent cubic (16) be employed, quadratic equations different from (17) and (18) are obtained, such, however, that their four roots are 2,, 1, 2, %,; but paired differently. It is therefore natural to expect that the three roots of (16) are (20) Yr=Uy%_ Ls q, Yg=UyX3 + XX, Yg=XyXy~ + Loz. It is shown in the next section that this inference is correct. 5. Without having recourse to Ferrari’s device, the two quad- ratic equations whose roots are the four roots of the general quartic equation (12) may be obtained by an @ priori study of the rational functions 2,%,+%,%, and x#,+%,—2,—2,=t. The three quantities (20) are the roots of (y—y,)(y—Yy2)(y—Yys3) =9, or C21) Yt Yet Ys)Ye + (YYot Vist YoY s)Y — YrYo's= 9. Its coefficients may be expressed * as rational functions of a,b,c, d: Yr t Yot Yg= Lye, + U3ly+ LL + Ll, +X,L,+2,X,=), MYT YrYst YoYg= — 40X20 324 + (Uy +2 q +3 + 4) (X,2Ly + LLC, + U,X3L, + L052) =ac—Ad, YYoYg= (L,Lyly + L,LyCy +L, LzL, + LyL_L,)" +204 § (Ly + Ly +%y +24)? —4(0,H+0,%_+...4+U50,)} =¢?+d(a?—4b), * This is due to the fact (shown in § 29, Ex. 2, and § 30) that any per- mutation of x,, %,, %, x, merely permutes ¥;, Y, 3, So that any symmetric function of y,, y., y; is asymmetric function of 2, x, x3, x, and hence rationally expressible in terms of a, , c. d. 8 GENERAL QUADRATIC, CUBIC, AND QUARTIC. [Cu. I Hence equation (21) is identical with the resolvent (16). Next, t? = (4, +2, +23 + 4)? — A(X, + %q)(%g +24) = 0? —A(a,0,+0,%,+ ... +20) +42,2,+475%, =a’?—4b+4y,. Again, 2,+2,+%7,+%2,=—a. Hence t,+%=t(t—a), %+2,=3(—t—a). To find x,x, and x,7,, we note that their sum is y,, while —t—a t—a —C=2,%,(%,+2,) +2,0,(2,+2,) =7,2, (==) tog, ) , ., U4%,=(c—fay,+ 4ty,)/t, T0,=(—e+ Zay,+ $ty,)/t. Hence 2, and 2, are the roots of (17), x, and x, are the roots of (18). 6. Lagrange’s a priorz solution of the quartic (12) is quite similar to the preceding. A root y,=2,7,+2,x, of the cubic (16) is first obtained. Then 2,7,=2, and x,7,=z, are the roots of 2?7—y z2+d=0. Then x,+2, and 73+, are found from the relations (%,+%,) + (X%3+2,) = —a, 2o(@, +22) +2, (ag + 2) = 2g Ly + Ugh Co FL Xots eg —az,+e az,—C '. YTn= eae &— &, &—&, ‘Hence x, and 2, are given by a quadratic, as also x, and a,. 7. In solving the auxiliary cubic (16), the first irrationality entering (see § 2) is A= (Y,— Yo) Yo— Ys) (Yi — Ys)- But Ya — Yo= (LZ —%4)(,— Ls), Yo—Yg= (4%, —X_)(Xz— Xs), Yr —Yg= (11 — Lp) (Lp — A), in view of (20). Hence (22) 4= (x, — L)(X, <4 Ty) (Ly or 4) (Lp ie Lg) (Xp —%,)(%3—2%,). SEC. 7] THEORY OF ALGEBRAIC EQUATIONS. By § 2, the reduced form of (16) is 7*+Py7+Q=0, where ee eA ely? (23) 1A ac—4d—4)?, Q=—a’d+ gabce+ $bd—c?— °,b%. Applying (8), with a change of sign, we get (24) 4=6V —3ViQ?+s,P%. CHAPTER II. SUBSTITUTIONS ; RATIONAL FUNCTIONS. 8. The operation which replaces 2, by ra, %, by Xp, 13 by 2;,..-, t, by t,, where a, &,..., vy form a permutation of 1) 2;..eeee is called a substitution on 2,,.%, %,,..., Yn. It is usually des- ignated Hct eedey MS hake Ta Uy Ly .-. Lyf] But the order of the columns is immaterial; the substitution may also be written To ie emer or (tn M1 2 Xs +e Ly La ty «+. Ly)’ Ly La Ly ty ...J?""” The substitution which leaves every letter unaltered, is called the identical substitution and is designated I. 9. THrorEM. The number of distinct substitutions on n letters is ni=n(n—1)...3-2-1. For, to every permutation of the n letters there corresponds a substitution. ExAmpuLeE. The 3!=6 substitutions on n=3 letters are: XT Ly Xz UT Ly 3 XU X_ Xs T= » a 4 hs b= , ’ X, Uz Lz U Xz X U3 XX Tin taate d Die Ley tg ae ae C= ; —- F c= . Ly a Ee 6 Py. A fo Ce Applying these substitutions to the function Y=2, + wt, +wx;, we obtain the following six distinct functions (cf. § 3); $1=2,+0%,+0'r,=$, $a=X,+ 0x, +72, =0%%, $b =X3+ wr, + wx, =v, 2 = , 2 ye = , Po=X,+ wr, + WX, Pa =%_+ WX, +W'X,=wW pe, He=X,+ Hr, + wr, =e, Io SEC. 10] THEORY OF ALGEBRAIC EQUATIONS. It Applying them to the function 6=(2x,—2,)(«,—2;)(x,—2,), we obtain $1 = fa= fo = 9, hc= ha = be = — . Hence ¢ remains unaltered by J, a, b, but is changed by e, d, e. 10. Product. Apply first a substitution s and afterwards a substitution t, where ae a eh ae ge ss. =. La yg... Me La Ly... Ly! The resulting permutation 2,’, Uy, + ++ 5 Ly’ CaN be obtained directly from the original permutation «,, 2,..., 2» by applying a single substitution, namely, um (2 Feed ee). Lat Ler «2. By! We say that wu is the product of s by ¢ and write w=st. Similarly, stv denotes the substitution w which arises by apply- ing first s, then ¢, and finally v, so that stu=uv=w. The order of applying the factors is from left to right.* Examp.es. For the substitutions on three letters (§ 9), ab=ba=I, ac=d;. ca=e, ad=e, da=e, aa=b, bb=a, abec=Ic=c, aca=da=c. Applying the substitution a to the function ¢, we get ¢a; applying the substitution c to ¢a, we get ¢a. Hence ¢ac=¢a. Likewise ¢an=¢1=¢, goa=¢. ; 11. Multiplication of substitutions is not commutative in general. | Thus, in the preceding example, ac¥ca, ad¥da. But ab=ba, so that a and 0 are said to be commutative. 12. Multiplication of substitutions is associative: st-v=s-tv. Let s, t, and their product st=wu have the notations of § 10. If by AG We ee ad Ee H by H 03 er acs pee. 8 Pa) ea Wet eo = (a2 O78 Sp Ha!’ af! eee Cyt! Xa 7 ee Ly Ss (2 --- Bn) _ (2, Ty ++. Fn) (Ta... \ ony 2 ” ’ ‘ ye ” © Lau Ln" Sad ay, La tp Raccoon Wg averse EXAMPLE. For 3 letters, ac-a=da=c, a-ca=ae=c. “* This is the modern use. The inverse order ts, vis was used by Cayley and Serret. [2 SUBSTITUTIONS ; RATIONAL FUNCTIONS. (Cu. II 13. Powers. We write s? for ss, s* for sss, ete. Then (25) Sgr = gmat (m and n positive integers). For, by the associative law, s™s"=s™.sst 1=smtign t= | | 14. Period. Since there is only a finite number n! of distinct substitutions on 7 letters, some of the powers Siybtn 8 wee eee must be equal, say s”=s™*", where m and n are positive integers. Then s™=s™s", in view of (25). Hence s” leaves unaltered each of the n letters, so that s"=T. The least positive integer o such that s’ =I is called the period of s. It follows that (26) 88 nies ee are all distinct; while s°t!, s°t?,..., s%*—!, 5% are repetitions of the substitutions (26). Hence the first o powers are repeated periodically in the infinite series of powers. ExampLes. From the example in § 10, we get a?=b, a’=a’a=ba=I, whence ais of period 3; b?=a, b3=b’%b>=ab=I, whence bis of period 3; c, d, e are of period 2; I is of period 1. 15. Inverse substitution. To every substitution s there corre- sponds one and only one substitution s’ such that ss’=J. If 3He . sy then sf=(%* fd ‘ La Up eee Ly aX, Xs ere Xn Evidently s’s=J. We call s’ the inverse of s and denote it hence- forun bys 9... tence SSan== 5 Sarde mn) es: If s is of period o, then s~'=s’—*'. Since s replaces a rational function f=/(%,,...,%n) by fs=/(®a,...,%v), 8~* replaces f, by f. EXxAMPLEs. For the substitutions on 3 letters (§ 9), a fA mt [La Ms ep) ns has Ve — (a v3 Hy) ster & U2 =) = vy a ai? oa PE =te™ naa OMe pe ye se fe bvleag, ori=¢, «dtd. foes anes [oe Src. 16] THEORY OF ALGEBRAIC EQUATIONS. 13 These results also follow from those of the examples in § 14. For the functions of § 9 the substitution a replaces ¢ by ¢a; a~!=b replaces ga by ¢. 16. THErorEM. Jf st=sr, then t=r. Multiplying st and sr on the left by s~1, we get Gest fn San Si 17. TororeM. If ts=rs, then t=r. 18. Abbreviated notation for substitutions. Substitutions like ess & Lo ah b= & Ls A), q= & Lz Xy ay a Pre 418 Tel rea which replace the first letter in the upper row by the second letter in the upper row, the second by the third letter in the upper row, and so on, finally, the last letter-of the upper row by the first letter of the upper row, are called circular substitutions or cycles. In- stead of the earlier double-row notation, we employ a single-row notation for cycles. Thus A=(X4Xp_X_), O=(XyzX4X), T= (L.T.7,T,). Evidently (2,225) = (Xo1'3%,) =(X%,2,), Since each replaces x, by X, L, by 3, and x, by x,. A cycle rs not altered by a cyclic permu- tation of its letters. Any substitution can be expressed as a product of circular substitutions affecting different letters. Thus Tie ©, Hy Va Ey Ls, Le\ _ ie L, :) = (%,)(2223), © Tg Le LyX, “) (©,Lg%5) (Log) (L,). A eycle of a single letter is usually suppressed, with the under- standing that a letter not expressed is unaltered by the substitution. Thus (2,)(a%3) 18 written (7,2). A circular substitution of two letters is called a transposition. 19. Tables of all substitutions on n letters, for n=3, 4, 5. For n=3, the 3!=6 substitutions are (compare § 9): T=identity, a=(2,7,7,), b=(x,2,2,), C= (x23), A=(X4%,), e=(X2,). 14 SUBSTITUTIONS ; RATIONAL FUNCTIONS. [Cu. Il For n=4, the 24 substitutions are (only the indices being written): I =identity; 6 transpositions: (12), (13), (14), (23), (24), (34); 8 cycles of 3 letters: (123), (132), (124), (142), (134), (1438), (234), (243) ; 6 cycles of 4 letters: (1234), (1248), (1324), (1342), (1423), (1432) ; 3 products of 2 transpositions: (12)(34), (13)(24), (14)(23). For n=5 the 5!=120 substitutions include I =identity; 5-4 oe. =10 transpositions of type (12); os =20 cycles of type (123); oes"? 30 cycles of type (1234) ; pee = 24 cycles of type (12345); 5-3=15 * products of type (12)(84); 20 + products of type (123)(45). EXERCISES. 1. The period of (123...) isn; its inverse is (nn—1...321). 2. The period of any substitution is the least common multiple of the periods of its cycles. Thus (123)(45) is of period 6. 3. Give the number of substitutions on 6 letters of each type. 4. Show that the function 2x,7,+2;x, is unaltered by the substitutions J, (2X), (1 3Xq), (Ly%2)(T3Xu), (XyXy) (yy), (1 pXq) (Loy), (TpLs% yy), (Ly yys). 5. Show that x, +32, is changed into 7,23 -+2,0, by (ps), (0,4), (%4%:%), (2,1 2X4), (XX Xy), (Lely), (Ly~yL4Ls), (1X32 4X2). 6. Write down the eight substitutions on four letters not given in Exs, 4 and 5, and show that each changes 7,2,+2 3%, Into 1,%,+2% 03. * Since the omitted letter may be any one of five, while one of the four chosen letters may be associated with any one of the other three letters. + The same number as of type (123), since (45) =(54). CHAPTER III. SUBSTITUTION GROUPS; RATIONAL FUNCTIONS. 20. A set of distinct substitutions s,, s,,..., Sm forms a group if the product of any two of them (whether equal or different) is a substitution of the set. The number m of distinct substitutions in a group jis called its order, the number n of letters operated on by its substitutions is called its degree. The group is designated Ge: All the n! substitutions on n letters form a group, called the symmetric group on n letters G”. In fact, the product of any two substitutions on n letters is a substitution on » letters. The name of this group is derived from the fact that its substitutions leave unaltered any rational symmetric function of the letters. ExAmpLeE 1. For the six substitutions on n=3 letters, given in § 9, the multiplication table is as follows: * Logon a. & fs Lee. Ue" ay e a Fy i gee Aa Oy: Ree » ese b els oi e,7d ¢ Cc Gi Oia. Did d de cee ened Les Db e ee Cape tiie.) Thus ad =e is given in the intersection of row a and column d. EXAMPLE 2. The substitutions J, a, b form a group with the multiplica- tion table to ent 4 7. i herrea te a ai a b ee a! * Tt was partially established in the example of § 10. 15 16 SUBSTITUTION GROUPS; RATIONAL FUNCTIONS. [Cu. Il If s is a substitution of period m, the substitutions LTS See See form a group of order m called a cyclic group. EXAMPLE 3. J, a=(123), b=a?=(132) form a cyclic group (Ex. 2). Examp.e 4, I, s=(123)(45), s?=(132), s?=(45), s*=(123), s°=(132)(45) form a cyclic group of order 6 and degree 5. 21. FUNDAMENTAL THEOREM. All the substitutions on ,, Yo, +++, Xn which leave unaltered a rational function P(x,, Lo, .-- , Ln) form a group G. Let $, denote the function obtained by applying to ¢ the sub- stitution s. Ifaand b are two substitutions which leave ¢ unaltered, then ¢.=¢%, ¢s=9-. Hence (pa)o=(P)o= Go=; or pab=9P- Hence the product ab is one of the substitutions which leave unaltered. Hence the set has the group property. The group G is called the group of the function ¢, while ¢ is said to belong to the group G. ExaAMPLeE 1. The only substitutions on 3 letters which leave unaltered the function (x,—2») (% —23)(%;—2,) are (by § 9) I, a=(a,%2%,), b=(x,7,2,). Hence they form a group (compare Ex. 2, § 20). Another function belonging to this group is (a, +wx,+w72,)?, wan imaginary cube root of unity. EXAMPLE 2. The only substitution on 3 letters which leaves unaltered x, +wx,+w*x, is the identity J ($9). Thus the substitution J alone forms a group G, of order 1. EXxampLe 3. The rational functions occurring in the solution of the quartic equation (§ 4) furnish the following substitution groups on four letters: a) The symmetric group G,, of all the substitutions on 4 letters. b) The group to which the function y, =2,2, + 23x, belongs (Exs. 4-6, p. 14): Gs={I, (12), (34), (12)(34), (13)(24), (14)(23), (1324), (1428)}. c) Since y,=2,%,+2,%, is derived from y,=27,2%7,+2%3%, by interchanging x, and xs, the group of y, is derived from G,; by interchanging x, and x; within its substitutions. Hence the group of y, is Gs’ = (1, (18), (24), (18)(24), (12)(84), (14)(32), (1284), (1432) }. SEc. 22] THEORY OF ALGEBRAIC EQUATIONS. 17 d) The group of y,;=2,%,+2,%,, derived from G, by interchanging z, and &,, 1S: G,” = (J, (14), (32), (14)(32), (13)(42), (12)(43), (1842), (1243) }. e) The function x,+2,—2x,—2, belongs to the group Since all the substitutions of H, are contained in the group Gy, H, is called a subgroup of G,. But H, is not a subgroup of G,’. jf) The function ¢=y,+wy,+w7y, or PH XyL, + U3lq + O(LyX3 + LyX) + w(L,X 4+ Lys), remains unaltered by the substitutions which leave y,, y,, and y, simulta- neously unaltered and by no other substitutions. Hence the group of ¢ is composed of the substitutions common to the three groups G;, G,’, G,”’, forming their greatest common subgroup: G,={I, r=(12)(34), s =(13) (24), ¢=(14)(23)} That these four substitutions form a group may be verified directly: phe og? rate taf rs=sr=t, rt=tr=s, st=ts=r. Hence anv two of its substitutions are commutative. This commutative group G, is therefore a subgroup of Gs, G,’, and G,”’. 22. THEOREM. Livery substitution can be expressed as a product of transpositions in various ways. Any substitution can be expressed as a product of cycles on different letters (§ 18). A single cycle on n letters can be expressed as a product of n—1 transpositions: (1234... n)=(12)(13)(14) .. . (In). Exampurs. ~(123)(456) =(12)(13)(45)(46), (132) =(13)(12) =(12)(23) =(12)(23)(45)(45). 23. THroREM. Oj the various decompositions of a given substi- tution s into a product of transpositions, all contain an even number of transpositions (whence s is called an even substitution), or all contain an odd number of transpositions (whence s 1s called an od substitution). 18 SUBSTITUTION GROUPS; RATIONAL FUNCTIONS. {Cu. UI A single transposition changes the sign of the alternating function * p=(x,—2X_)(@, —Xg)(%, — 2X4) see (1,—2n) slp rahe) (tte 2) came (X,—2Xn) e . e e . Gan Besa Thus (,2,) affects only the terms in the first and second lines of the product, and replaces them by (1 —2X,)(L_— Xz) (,— Ly) . . . (Ly— Ap) (1, —%3)(2,— 4%)... (Hae Hence, if s is the product of an even number of transpositions, it leaves ¢ unaltered; if s is the product of an odd number of trans- positions, it changes ¢ into —¢. CoroLuaRyY. The totality of even substitutions on 7 letters ° forms a group, called the alternating group on n letters. EXAMPLE 1. The alternating group on 3 letters is ($§ 9, 19) G3) = tis (123), (132) }. EXAMPLE 2. The alternating group on 4 letters is (§ 19) Gy,(4) = {I, (12)(34), (13)(24), (14)(23), and the 8 cycles of three letters}. 24. THrorEeM. The order of the alternating group on n letters is 4-n! Denote the distinct even substitutions by (e) C1) Co, Czy 2 + +» Cke- Let ¢ be a transposition. Then the products (0) Biber Cal, Went ts tay Cee are all distinct (§ 17) and being odd are all different from the substitutions (e). Moreover, every odd substitution s occurs in * It may be expressed as the determinant MA A eS a 2 —1 Le ety ees Saekee e . e * . . aa Ltn fen ee ie ee Src. 25] THEORY OF ALGEBRAIC EQUATIONS. 19 the set (0), since st is even and hence identical with a certain e;, so that Se Cian 0,1. Hence the 2k substitutions given by (e) and (0) furnish all the n! substitutions on n letters without repetitions. Hence k=}-n! 20. As shown in § 21, every rational function d(a7,,..., Xn) belongs to a certain group G of substitutions on 2,,..., Z,, namely, is unaltered by the substitutions of G and changed by all other substitutions on 2,,...,2%,. We next prove the inverse theorem: Given a group G of substitutions on x,,...,%n, we can construct a rational function f(2,,...,Xn) belonging to G. Let G= {a=I, b,c,..., 1} and consider the function V=M,2,4+ Mot, + ... +MnXn, where m,, m,,..., M» are all distinct. Then V is an n!-valued function. Applying to V the substitutions of G, we get (27) eS | el eee ae ay all of which are distinct. Applying to (27) any substitution c of G, we get (28) Wren eV Le These values are a pertiutation of the values (27). since ac, bc,... , Ic all belong to the group G and are all distinct (§ 17). Hence any symmetric function of V,, Vy,..., V; 1s unaltered by all the substitutions of G. By suitable choice of the parameter p, the . symmetric function p=(0—V)(o—Ve)(o—V-) --. (e—Vi) will be altered by every substitution s not in G. Indeed, p= (p— Vie)(o— Vis)(o— Vicelenet (o— Ve} is not identical with ¢ since V, is different from V, Vs, V.,...+ Vi. ExamplLeE 1. For G={I, a= (14,25), b=(a,2,7,)}, take V =2,+07,+ 0%. Then Va=w?V, Vp=wV. Hence VtVatVo=(1+0+0)V=0, VVa+VVs+VaVo=0, VVaVo=V". The function V* belongs to G (see Ex. 1, § 21). - 20 SUBSTITUTION GROUPS ; RATIONAL FUNCTIONS. (Cu. II EXAMPLE 2. For G={I, c=(a,%,)}, take the V of Ex.1. Then VVc=(4, +02, + 7x,)(%,+w2,+72,) =c,?—3c, is unaltered by all six substitutions on the three letters. But b=(e—V)(o—Ve) =p? — (2%, — 2, —23)9 + Cy? —3e, for 00, is changed by every substitution on the letters not in G. Hence, for any 0+0, ¢ belongs to G. EXERCISES Ex. 1. If wisa primitive uth root of unity, (%, +02, +07%3+ ... +wh—tey)4 belongs to the cyclic group {J, a, a?,..., a*—'}, where a=(a, 4%... 2p). Ex. 2. Taking V =2,+7%,—z,—1a, and s=(x,%.)(%,%,), show that VV s=i(x,—2,)?+i(2,—2,)? belongs to G, of § 21, that V+Ve belongs to H, of § 21, while (o—V)(o—Vs), for 040, belongs to theegroup {J, s} Ex 3. Taking V=a,+7ix,—2x,—ix, and t=(2,23)(2,2,), show that VV; — belongs to the group {J, ¢} Ex. 4. If a,, a,. ., an are any distinct numbers, the function V =2,%1 X,%2 - Ln in is n!-valued, and V+ V,+Vc+ ...+V: belongs to {I,b,c,. .,l}. Ex. 5. If é belongs to G and ¢’ belongs to G’, constants a and a’ exist such that af+a’¢’ belongs to the greatest common subgroup of G and G’, 26. THroreM. The order of a subgroup ts a divisor of the order c) the group. Consider a group G of order N and a subgroup H composed of the substitutions (29) heels dudes tise. If G contains no further substitutions. N=P, and the theorem is true. Let next G contain a substitution g, not in H. Then G contains the products (30) Gas RaGos MeGay-- any LEQo- The latter are all distinct (§ 17), and all different from the sub- stitutions (29), since hag,=hg requires that g,=ha'hg=a sub- Sec. 27] THEORY OF ALGEBRAIC EQUATIONS. 21 stitution of H contrary to hypothesis. Hence the substitutions (29) and (80) give 2P distinct substitutions of G. If there are no other substitutions in G, N=2P and the theorem is true. Let next G contain a substitution g, not in one of the sets (29) and (30). Then G contains (31) Js) RoGa: heJs,- - - » APY; As before, the substitutions (31) are all distinct and all different from the substitutions (29). Moreover, they are all different from the substitutions (30), since hag,=hsg, requires that g,=hi'heg, shall belong to the set (30), contrary to hypothesis. We now have 3P distinct substitutions of G. Either N=3P or else G contains a substitution g, not in one of the sets (29), (80). (31). In the latter case, G contains the products (32) Oi Wels eae amd all of which are distinct and all different from the substitutions (29), (30), (31), so that we have 4P distinct substitutions. Pro- ceeding in this way, we finally reach a last set of P substitutions (33) Jv, h.gv, hey; RTO} hpq., since the order of H is finite (§ 9). Hence N=vP. DEFINITION. The number =>, is called the index of G Y the subgroup H under G, and the relation is exhibited in I the adjacent scheme. CoroLuaRy. The order of any group H of substitutions on n letters is a divisor of n! Indeed H is a subgroup of the symmetric group G,, on n letters. 27. THrorremM. The period of any substitution contained in a group G of order N is a divisor of N. If the group @ contains a substitution s of period P, it contains the cyclic subgroup H of order P: iteet Bestar Spr kt Slot Then, by § 26, P is a divisor of N. 22 SUBSTITUTION GROUPS ; RATIONAL FUNCTIONS. [Cu. Il CoroLLARyY.* If the order N of a group G is a prime number, G is a cyclic group composed of the first N powers of a substitution — of period N. 28. As shown in § 26, the N substitutions of a group G@ can be arranged in a rectangular array with the substitutions of any subgroup # in the first row: Wee Palace Wicks, aa ee Jo Mog. Nyy .-- hrge Js hogs ggg --- Pgs gv hogy hego .-- beg r Here g,=/, 9, 93,+++, 9» are called the right-hand multipliers. They may be chosen in various ways: g, is any substitution of G not in the first row; g, any substitution of G not in the first and second rows; g, any substitution of G not in the first, second, and third rows; ete. Similarly, a rectangular array for the substitutions of G may be formed by employing left-hand multipliers. 29. THrorEM. Ij ¢ is a rational function of x,...., Ln belonging to a subgroup H of index v under G, then ¢ is v-valued under G. Apply to ¢ all the N substitutions of G arranged in a rect- angular array, as in § 28. All the substitutions belonging to a ~ row give the same value since Prida - (Dn, aoe as (P),, =. Pons Hence there result at most v values. But, if Yo.= a, (BWe 27° Initially given are the elementary symmetric functions Yt YotYs=9, YYotYVYstYYs=P, —YWYYs=4%, belonging to the symmetric group G, on Y,, Yo, Y3- Solving a quadratic resolvent equation, we find the two-valued function &, which belongs to the subgroup G, of G, (§ 21, Ex. 1). Solving next a cubic resolvent equation, we find the six-valued function 2, which belongs to the subgroup G, of G, (§ 21, Ex. 2). Then y,, y, ys are rational functions of z, p, g, since they belong to the respective groups Ge’ = (1, (Yos)}, Go’ = tL, (Vis) $s Ge!” = 11, (Wye) $5 each containing G, (also direct from § 381, Cor. 2). From the group standpoint, the solution is therefore expressed by the scheme: Ge: DP, q 2 | Ag ee 7 porns Gel Yg G22 Ga:2 Gita Giz 27 28 GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. IV 34, The same method leads to a solution of the general cubic x? —c,x? +¢,7 —¢,=0. To the symmetric group G, on 2,, 2, 7, belong the functions Lz +Xo+Le=Cy, LiX_etXzXyetLX,—=Cy, U,HoXy=Ceg. To the subgroup G,= {J, (2,224), (2,%%)} belongs the function A= (X,—X2)(X_— Xg)(X3— 2). In view of Ex. 3, page 4, 4 is a root of the binomial resolvent 4? =¢,7c,” + 18¢, CC, — 4,3 — 4¢,3¢, — 27c,?. By § 3 and § 2, we have for ¢,=2,+w2,+w2,. $4=2,+wt,+ws, dg it+t¢ f= 2¢7%—9c,c,+27c,, by pie BV —3(2, — Hq) (L_— Lg) (Lg = 2) = — 8V—3 4. *, b 8=4(2c,3—9c,c,+27¢,—3V —3 4), $F=4(2c,3—9c,c,+27¢,+3V —3 4). After determining * ¢, by extracting a cube root, the value of dy is (§ 3) , Jy = (C1? —3¢,) + gy. Then, as in § 3, 2, 7, x, are rationally expressible in terms of ¢,: BH=Uaqthtys), 2=3(4, tod, +w9,), %=4F(¢,+wg,+wg,). 35. The solution given in § 5 of the general quartic equation (12) x*+ax?+ bx? +cx+d=0 may be exhibited from the group standpoint by the scheme: CPR OR! | ae 2% 2 fy SY, = 2X, +0, V(t, +a, 7, ap TT 36, Beda We ee Ue Here H,= {I, (,2,)}, H,’={I, (a,2,)}, G, and H, being given in § 21. * For another method see Ex. 4, page 41. Sxc. 36] THEORY OF ALGEBRAIC EQUATIONS. 20 36. Lagrange’s second solution of (12) is based upon the direct computation of the function 2,+2,—2x,—.,. Its six conjugate values under G,, are +¢,, +¢,, +t, where Q=X,t%,—My—-T, = X%,+%3—X,—X%, t3=X,+X,—X_,—y. The resolvent sextic is therefore (2? —1,?)(7? —1,7)( 7? —#,7) =0. Its coefficients may be computed * easily by observing that t’?=a?—4b+4y,, t,2=a°—4b+4y,, t,?=a?—4b+4y,, as follows from § 5. Using the results there established, we get ty? +t,’+ t,? = 3a? —12b+4 4(y,+ y,4+ y,) =3a?—8b, t,7t,?+ t,7t,’ + t,t,” = 3(a —4b)?+ 8(a? —4b)(y,+ y+ Yo) + 16(Y:Yo+ YiYat Yos) = 3a*— 16a7b+ 16b?+ 16ac— 64d, t,t,7t,” = (a? Beto) + 4(a? z 4b)*(y, T Yet Ys) + 16(a* — 4b) (YYot YiYst Yes) + 649, YoYs = {8c+ a(a?—4b)}?. The resolvent becomes a cubic equation upon setting t?=o. De- note its roots by o,=1,’, o,=t,’, o,=t,’. Then L4+ t,—X,—2,=V 6, £,+2,—21,—2,=V 0, Uy X, —%,—2z=V/G,, 2+ L,+ t+ 2,= —4. From these we get m= —at+Vo,4Vo,4Vo,), =H —a+Vo,—Va,-Va,), t= 4( —A—W/G,4-V/o,—Vo,), 2,=1(—a—V0,—Va,+ V9) The signs of V/ o, and Vo, may be chosen arbitrarily, while tha‘ «| of V 0, follows from (39) V6iV6,V 6, = ttt, = 4ab —8e—a’. Indeed, we may determine the sign in t,t,t,—= + {8c+ a(a’—4b)} * Compare Ex. 5, page 41. 30 GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. IV by taking z,=1, 7,=27,=2,=0, whence a= —1, b=c=d=0, t,t,t,=1. 37. The following solution of the quartic is of greater interest as it leads directly to a 24-valued furiction V, in terms of which all the roots are expressed rationally. As in § 5, we determine y, and t, belonging to G, and H, respectively, by solving a cubie and a quadratic equation. To the subgroup G= (1, (Xy%,)(_,) } of H, belongs the function ¢=V?, while to G, belongs V, where V=(2,—2,)+ ua, —2,)- Under H,, ¢ takes a second value ¢,={(2,—2%,)—(7,—2,)}?. Then z’—($+ ¢,)e+ $f,=0 is the resolvent equation for ¢. But a= (x —2,)?+ (@—2,)"}? = {a? — 2b -2y, }?= 4480? 8b, Ut (y= 24) a, = er) 2 = 2(%,—2%,+ L,—X,)(@, —Ty eee = 2(4ab—8c— a?) +t, in view of (39). After finding ¢~ and ¢,, we get VaV¢. Vi=Vb,=(a,—2,) -(a,—2,); (40) V,=4(3a?—8b—?)+V. Having the four functions t, V, V,, and 7,+.2,+2%,+%,=—a, we get m=H—att+V+V,), 2%=+(—a+t-V-V)), Ot) med =a tt) ee 38. The solution of the general cubic (§ 34) and the solution of the general quartic (§ 37) each consists essentially in finding the value of a function which is altered by every substitution on the roots and which therefore belongs to the identity group G,. Like- wise, the general equation of degree n, (42) 2 OTN Coe ta one om LG could be completely solved if we could determine one value ofa function belonging to the group G,; for example, Src. 38] THEORY ©OF “ALGEBRAIC EQUATIONS. 31 (43) V=mM,2,tMytot+ ... +MnXn (m’s all distinct). In fact, each x; 1s a rational function of V,c¢,,...,¢,n by § 31. For the cubic and quartic, the scheme for determining such a function V was as follows: Pires, vol, Cy Go. Grand, 2 | 3 | G,: (2%, +wx,+w?2,)% Gil Filet Pel, 3 | 2 | G,:2,twt,+ wr, A 4:%,+%,—4,—%, 9 Gy: (2, — 1p +-12%,—12,)? 2 | G,:%,—X_ +14, —12, The same plan of solution applied to (42) gives the following scheme: Pitas Cay eee 7on d | (Sede ered C eds Cae a teas te =O) ue | ‘Sep elitty sons Ca) TC otis sO M:¢ a | Cave VOI Oe Cit.) Ga i... =O. Such resolvent equations would exist in view of the theorem of § 32. In case the resolvent equations were all binomial, the function V (and hence g,,..., X,,.) would be found by the extraction of roots of known quantities, so that the equation would be solvable by radicals. We may limit the discussion to binomial equations of prime degree, since z?¢=A may be replaced by the chain of equations z?=u, w4=A. The following question therefore arises: G: If vy | , when will the resolvent equation for / take the form i: 34 (44) gv=Rat. Func. (¢,¢,- «5 Cp). 32, GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. IV Since v is assumed to be prime, there exists a primitive vth root of unity, namely a number w having the properties w’=1, wk] for any positive integer k< v. Hence the roots of (44) may be written (45) CeO, Us ere same Let ¢,=¥¢, ¢2,..., Yy denote the conjugate functions to ¢ under G (their number is v by § 29). Now ¢ belongs to the group H by hypothesis. Let ¢, belong to the group H,, ¢, to Hs, ..., g, to Hy. Since the roots (45) differ only by constant factors, they belong to the same group. Hence a necessary condition is that Ho eee 39. The first problem is to determine the group to which belongs the function ¢, into which ¢ is changed by a substitution s, when it 1s given that ¢ belongs to the group - HM eth=lo Io ee : If a substitution o leaves ¢, unaltered, so that ¢,,= qs, then ie oes pee Pss-t = i. Hence sos~!=h, where h is a substitution of H. Then o=s" hs. Inversely, every substitution s~hs leaves ds unalterel. Hence ds belongs to the group 187 has =], 8 saligs 2 eee ee aoe which will be designated s~'Hs. We may state the theorem: If ¢ belongs to the subgroup H of index v under G, the conjugates ¢, $955 ae a) Jar of } under G, belong to the respective groups _ Tp SOse rg gag kOe. LL Oae DeFrinitions. ‘The latter groups are said to form a set of con- jugate subgroups of G. In ease they are all identical, H is called a self-conjugate subgroup of G (or an invariant subgroup of G). Src. 40] THEORY OF ALGEBRAIC EQUATIONS. 33 Hence a necessary condition that the general equation of degree n shall be solvable by radicals under the plan of solution proposed in § 38 is that each group in the series shall be a self- conjugate subgroup of prime index under the preceding group. Note that the group G,={/} is self-conjugate under every group G since g-'Ig=I. EXAMPLE 1. Let G be the symmetric group G, on 3 letters and let H be the group G,;={J, (2,273), (a@,%2,)}. Let g,=(x,7;). Then $ =(2,+0r,+*2,)?, $o, = (2+ wx, + w2;)* form a set of conjugate functions under G. Now ¢ belongs to H and ¢, belongs to the group {J, (a,7,2,), (@,2,7,)}, whose substitutions are derived from those of H by interchanging the letters x, and 2,, since that interchange replaces ¢ by ¢g,. To proceed by the general method, we would compute (20s) (L223) (Lys) =(HyLsCp), (W_Xy) —'(XyL3Xp) (L_T,) = (TT). By either method we find that the group of ¢ and ¢g, are identical, so that G, is self-conjugate under G, Also, G, is self-conjugate under G,. Hence the necessary condition that the general cubic shall be solvable by radicals is satisfied EXAMPLE 2. Consider the conjugate values 2, 2, x, of x, under G,: i (223) | Jz =(X,%q), (2X3) Jo = (242,75) | Te 93=(2y25), (X_X2) 93 =(XyX2%y) | Lz Hence H = {J, (x,7;)} is not self-conjugate under G,. Here 9, 'Hg, = (I, (a ,23) } ~H, 93 'Hg3= if, (x,22) ; “HH, 40. Derinitions. Two substitutions a and a’ of a group G are called conjugate under ( if there exists a substitution g belong- ing to G such that g~'ag=a’. Then a’ is called the transform of a by g. There is a simple method of finding g~'ag without performing the actual multiplication. Suppose first that a is a circular sub- stitution, say a=(af70), while g is any substitution, say 34 GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. 1V Hence g~'!ag=(a’f’y’0’) may be obtained by applying the substi- tution g to the letters of the cycle a=(afy0). Let next a=a,a,a,..., Where a,, d,... are circular substitu- tions. Then . g—ag=g~'a,9-9~'dog-G "dg... Hence g~ag is obtained by applying g within the eyeles of a. Thus (123)-".(12)(34) (123) =(23)(14). CoROLLARY. Since any substitution transforms an even sub- stitution into an even substitution, the alternating group Gj, 1s a self-conjugate subgroup of the symmetric group Gy). 41. Turorem. Of the following groups on four letters: Gy, Gy, Gy= 11, (12)(34), (13)(24), (14)(23)}, 6,= {TDG Gea each is a self-conjugate subgroup oj the preceding group. By the Corollary of $ 40, G;,, is self-conjugate under G.,. To show that G, is self-conjugate under G,, (as well as under G,,), we observe that G, contains all the substitutions of the type (a@f)(70), while the latter is transformed into a substitution of the form (a’3’)(7'9") by any given substitution on four letters. That G, is self-conjugate under G, follows from the fact that (12)(34), (13)(24), (14)(23) all transform (12)(34) into itself.* 42. The necessary condition ($39) that the general quartic x'+anr*+ ba*+cx+d=0 shall be solvable by radicals is satisfied in view of the preceding theorem. We proceed to determine a chain of binomial resolvent equations of prime degree which leads to a 24-valued function V=2,—2,4+1%,—12,, * This also follows from § 21, Ex. (/), since rs=sr gives s—'rs =r. Suc. 42] THEORY OF ALGEBRAIC EQUATIONS. 35 in terms of which the roots 2,, X., 3, %, are rationally expressible. Let (20) YAU Lo A Lely, Yo=HUXyLgt Uy, Yg= ot hg, as in $4. The scheme for the solution is the following: Pri axon c,d Gy d =(X,—22)(H,—Xg)(@y — 4) (Lp — Vy) (Lp — Ly) (Lz — 4) ae 1 Dy=Y, FWY. + wy; Ae 2A = G+ (4, +2,—23—2,) 2 G,:V =2,—2,+1%,—12, Referring to formule (22), (23), (24) of § 7, and setting P= —4J, Q=16/, we get 4=16V P—27J?, er ead alo 8 he tel ot ri Ge 1G 216; Hence 4 is a root of the binomial resolvent 4?=256(/*—27J’). The resolvent for ¢, is the binomial equation (p—9$1)(P—wg,)($—-07h,) =P? — 9° =0. By Lagrange’s Theorem, ¢,° is a rational function of 4, a, b, c, d. To determine this function, set d,=Yy, +w?Y,+ wy. Then ($§ 2, 7) $3—$2=38V —3(Y,—Y2)(Yo—Ys) (Ys Ys) = —8V —3 4 fet by =2(y2+Yy2+Ys°) + L2y,YoY3+3(w+w")d, where O=Y,"Yo+YYo + YY: +YYs +Y2°Yst+YoYs° Satisfies the rela- tions (Yr tYot Ys) (YiYo+ YY s+ Yols) = 9+ 38Y:YoYss (Y; + Yo+ Ys)? = 30 + BY, YoYg t+ Yy>+ Yn? + Y;°. . Po FG F=2(y, + Yt Ys)>— WY + Yot Ys(YiYot YiYst Yos) + 27y,YoYs = 2b*— 9b(ac—4d) +27(c?+a2d—4bd) = — 432 36 GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. IV upon applying the relations in § 5. Hence Jb 2=413V —34—216J. In view of Lagrange’s Theorem, y,, y,, and y, are rational functions of ¢, These functions may be determined as follows: ONS a che Yo Ya. +(w +w?)(Y,Yo + Y1Y3 + YxYz) =(Y,+Y2+ Ys)? — 3(Y1Y2+ YrYs + YoYs) =6*—3ac+12d=H. H From Y%t+YotYys=b, YyHWY.t+w7ys=hy, Yy+w7Y,+wYs= = et n= 4(b+$.47), w= a(d-+0%g.+), n= 4(d+op. +). Setting t=2,+7,—2,—2,, we obtain for A=¢,/t the binomial resolvent P=,’ + (a? —46+4y,), upon replacing ?? by its value given in § 5. Next, we have (§ 37) V?= (1-2)? — (&g— 24)? + 20(4 — Lp) (Ly — Ly) 4ab—S8c—a? = eae oe e + 22(4fo— Ys) = -(4ab—80—a") +3V3($,-7). The values of 2, 2, Ys, 2, are then given by (41) in connection with (40). SERIES OF COMPOSITION OF THE SYMMETRIC GROUP ON n LETTERS. 43. Drrinitions. Let a given group G have a maximal self- conjugate subgroup H, namely, a self-conjugate subgroup of G which is not contained in a larger self-conjugate subgroup of G. Let H have a maximal self-conjugate subgroup K. Such a series of groups, terminating with the identity group G,, G) He Ken, ee Src. 44] THEORY OF ALGEBRAIC EQUATIONS. Rf in which each group is a maximal self-conjugate subgroup of the preceding group, forms a series of composition of G. The num- bers A (the index of H under G), » (the index of A under H),..., o (the index of G, under M) are called the factors of composition of G. If the series is composed of the groups G@ and G, alone, the group G is called simple. ‘Thus a simple group is one containing no self-conjugate subgroup other than itself and the identity group. A group which is not simple is called a composite group. ExampteE 1. Fo she symmetric group on 3 letters, a series of composition is G,, G3, G, (see Ex. 1, § 39). Since the indices 2, 3 are prime numbers, the self-conjugate subgroups are maximal (see § 26). EXAMPLE 2. A series of composition of the symmetric group on 4 letters is Gy, Gy, G,, G., G, (§ 41), the indices being prime numbers. EXAMPLE 3. A cyclic group of prime order is a simple group (§ 26). 44, Lemma. Jf a group on n letters contains all circular sub- stitutions on 3 of the n letters, vt is ether the symmetric group Gn, or else the alternating group Gy). It is required to show that every even substitution s can be expressed as a product of circular substitutions on 3 letters. Let s=hl, eS a boy —sloy where ¢,,...,¢, are transpositions (§$§ 22, 23), and 4,42. If 4 and t, have one letter in common, then tity = (a8) (ay) =(a67). _If, however, ¢, and ¢, have no letter in common, then tyt,= (a3)(70) = (a9) (ar)(7a)(79) = (487) (a0). Similarly, i,t, is either the identity or else equivalent to one cycle on 3 letters or to a product of two such cycles. Hence the group contains all even substitutions on the n letters. 45. THrorem. The symmetric group on n>A4 letters contains no selj-conjugate subgroup besides itself, the identity G,, and the alternating group Gin, so that the latter is the only maximal self- conjugate subgroup of Gy) (n>A4). 38 GENERAL EQUATION FROM THE GROUP STANDPOINT. |Cu. IV That the alternating group is self-conJugate under the symmetric _ group was shown in § 40. Let G,, have a self-conjugate Se eee H which contains a substitution s not the identity J. Suppose first that s contains cycles of more than 2 letters: s=(aheU aa) jc eee Let a, 8, 0 be any three of the n letters and 7, «,..., o,... the remaining n—3 letters. Then H contains the substitutions = (aby... dEb..)..., = Gar... ES...) .05, the letters indicated by dots in s, being the same as the correspond- ing letters in s,. The fact that s, (and likewise s,) belongs to H follows since pa (tOc... def... Na Poy). 23 BO one is a substitution on the n letters which transforms s into s, (§ 40), while any substitution o of G,, transforms a substitution s of the selj-conjugate subgroup H into a substitution belonging to H (§ 39). Since H is a group, it contains the product s,s,~!, which reduces to (afd). Hence H contains a circular substitution on 3 letters chosen arbitrarily from the n letters. Hence ZH is either Gn Or Gyn) (§ 44). Suppose next that s contains only transpositions and at least two transpositions. The case s=(ab)(ac)...=(abc)... has been treated. Let therefore s=(ab)(cd)(ef) .. . (Im). Let a, 2, 7, 0 be any four of the n letters, and ¢, ¢d,..., A, w the others. ‘hen the self-conjugate subgroup H contains the sub- stitutions =(aB)(70)(eP) ..- CA), 8:= (ar) (G0) (ed) «. . (Aw) and therefore also the product s,s,~', which reduces to (ad)(By). Src. 46] THEORY OF ALGEBRAIC EQUATIONS. 39 Since n>4, there is a letter o different from a, 2, 7, 0. Hence H contains (a)($7) and therefore the product (ad) (87) -(ae)(G7) = (ade). It follows as before that H is either Gy; or Gyn). Suppcse finally that s=(ab). Then the self-conjugate subgroup H contains evcry transposition, so that H=G,,.. 46. THrorem. The alternating group on n>4 letters is simple. Let G:,: have a self-conjugate subgroup H larger than the identity group G,. Of the substitutions of H different from the identical substitution /, consider those which affect the least number of letters. All the cycles of any one of them must contain the same number cf letters; otherwise a suitable power would affect fewer letters withcut reducing to the identity J. Again, none of these substitutions contains more than 3 letters in any cycle. Tor, if H contains pers ee ayn) then H contains its transform by the even substitution o= (234): ieee So atv 0) ie). oe; where the dots indicate the same letters as in s. Hence H would contain gs tel a2) affecting fewer letters than does s. Finally, none of the substi- tutions in question contain more than a single cycle. For, if H - contains either ¢ or s, where bat oA). , S—(123)(456)......, it would contain the transform of one of them by the even substi- tution x=(125) and consequently either ¢-«~'tx or s7!-x7'1sx. The latter leaves 4 unaltered and affects no letter not contained in s; the former leaves 3 and 4 unaltered and affects but a single letter 5 not contained in ¢t. In either case, there would be a reduc- tion in the number of letters affected. The substitutions, different from J, which affect the least num- ber of letters are therefore of one cf the types (ab), (abc). The former is excluded as it is odd. Hence H contains a substitution 40 GENERAL EQUATION FROM THE GROUP STANDPOINT. [Cu. IV (abc). Let a, 8, 7 be any three of the n letters, 0, ¢,..., v the others. Then (abc) is transformed into (afr) by erther of the substitutions al ae oe 8) s= (0 eee GO BaD See ye af yoo pears where the dots in 7 indicate the same letters asin s. Since r=s(de), one of the substitutions 7, s is even and hence in Gy. Hence, for n>4, H contains all the circular substitutions on 3 of the n letters, so that H=Gj,). 47. It follows from the two preceding theorems that, for n>4, there 1s a single series of composition of the symmetric group on n letters: Gn:, Gin}, G,. The theorem holds also for n=8, since the only subgroup of G, of order 3 is G,, while the three subgroups of G, of order 2 are not self-conjugate (§ 39, Ex. 2). The case n=4 is exceptional, since G,, contains the self-conjugate subgroup G, (§ 41). Except for n=4, the factors of composition of the symmetric group on n letters are 2 and 3n!. 48. It was proposed in § 38 to solve the general equation of degree n by means of a chain of binomial resolvent equations of prime degrees such that a root of each is expressible as a rational function of the roots 2,, %,..., %, of that general equation. As shown in §§ 38-389, a necessary condition is the existence of a series of groups (46) Gin, aK ee eines each a self-conjugate subgroup of prime index under the preceding group. In the language of § 43, this condition requires that Gp; shall have a series of composition (46) with the factors of com- position all prime. By § 47, this condition is not satisfied if n}5, since 4n!is then not prime. But the condition is satisfied if n=3 or if n=4 (§ 39, Ex. 1; § 41). Under the proposed plan cf solu- tion, the general equation of degree n>4 is therefore not solvable by radicals, whereas the general cubic and general quartic equa- tions are solvable by radicals under this plan (§ 34, § 42). Src. 48] THEORY OF ALGEBRAIC EQUATIONS. 41 To complete the proof of the impossibility of the solution by radicals of the general equation of degree n>4, it remains to show that the proposed plan is the only possible method. This* was done by Abel (@uwvres, vol. 1, page 66) in 1826 by means of the theorem : Every equation which rs solvable by radicals can be reduced to a chain of binomial equations of prime degrees whose roots are rational functions of the roots of the given equation. As the direct proof of this proposition from our present stand- point is quite lengthy, it will be deferred to Part II (see § 94), where a proof is given in connection with the more general theory due to Galois. EXERCISES. 1. If H={I, h,... , hp} is a subgroup of G of index 2, H is self-conjugate under G. Hint: The substitutions of G not in H may be written g, gh,,..., ghp} or also g, hyg,..., hpg. Hence every hgg is some gha, so that for every ha, g—‘hgg is some ha. 2. The group G, of § 21 has the self-conjugate subgroups G,, G,, H,, C,={I, (1824), (12)(34), (1423)}. The only remaining self-conjugate sub- groups are G, and G. 3. If a group contains all the circular substitutions on m+2 letters, it contains all the circular substitutions on m letters. Hint: (123...mm4+1m-+2)(mm-1...382m4+21m+1)=(123...m—1 Mm). 4, Compute directly the function ¢,° of § 34 as follows: Py =13 +2 y3 +24) + Ox yw Ly + 80(xy7X, +1 4X4" +.%yX) + 3w?(Xyxy? +0y7x; +13”) = 2,3 +223 + 25° + Ox ,Lyes — 3 (Nyy + LyLy? +0473 + WyLy” + L273 + Lys” —3Vv —34, since 1? —LyLq? +4424” — 11+ 12°, — TL” = — (1-22) (#2 —2s) (%—2,) =—4, Twice the remaining part of ¢,3 equals 2c,3—9c,c,+27c, by § 3. 5. Compute directly the coefficients in § 36 as follows: t?+t.?+t,? =3 2 2;? —2 5x7; =3a?—8b, Lily = DU 2) Ly lok, — 22 ( Xo +-2,° + 2,7) =227,34+222,0,%,— 24; 2;? =4ab —8c—a? * For the simpler demonstration by Wantzel, see Serret, Algébre, II, 4th or 5th Edition, p. 512, SECOND PART. GALOIS’ THEORY OF ALGEBRAIC EQUATIONS. CHAPTER VY. ALGEBRAIC INTRODUCTION TO GALOIS’ THEORY. 49. Differences between Lagrange’s and Galois’ Theories. Here- tofore we have been considering with Lagrange the general equation of degree n, that is, an equation with independent variables as coefficients and hence (see page 101) with independent quantities L,; Uye++, ty aS roots. Hence we have called two rational functions of the roots equal only when they are identical for all sets of values of 2,,..., %n. But for an equation whose roots are definite constants, we must consider two rational functions of the roots to be equal when their numerical values are equal, and it may happen that two functions of different form have the same numerical value. Thus the roots of x3+27?+2+1=0 are “,=—1, ~=+1, %=-1 (=Vearp Hence the functions 2,’, x,’, and x, are numerically equal although of different form. We may not apply to the equation 2x,?=2,? the substitution (7,77), since 2,?2,”. Again, the totality of the substitutions on the roots which leave the function z,? numerically unaltered do not form a group, since the substitutions are J, (2,23), (X05), (4X22). 42 Suc. 50] THEORY OF ALGEBRAIC EQUATIONS. 43 Again, the reots of r*+1=0 are w= 6, =, %——é, = —te (=). /2, Hence v,?=e?=1, 2,2,=¢«7=1. The functions xz,’ and 2,2, differ in form, but are equal numerically. Also, 2,? equals 2,4”, but differs from 2,” and x,°._ The 12 substitutions which leave 2,? numerically unaltered are ,(23),(24),(34),(234) ,(243),(13),(13)(24),(213),(413), (4213), (4182), the first six leaving x,’ formally unaltered and the last six replacing z,? by x,’.. They do not form a group, since the product (13)(23) is not one of the set. There are consequently essential difficulties in passing from the theory of the general equation to that of special equations. This important step was made by Galois.* In rebuilding our theory, special attention must be given to the nature of the coefficients of the equation under discussion, (1) a” —¢,2"-14¢,4"—-*7— ... +(—1)"c,=0. Here ¢,,...,¢€n may be definite constants, or independent variables, or rational functions of other variables. Whereas, in the Lagrange theory, roots of unity and other constants were employed without special notice being taken, in the Galois theory, particular attention is paid to the nature of all new constants introduced. 50. Domain of Rationality. To specify accurately what shall be understood to be a solution to a given problem, we must. - state the nature of the quantities to be allowed to appear in the solution. For example, we may demand as a solution a real num- * fivariste Galois was killed in a duel in 1832 at the age of 21. His chief memoir was rejected by the French Academy as lacking rigorous proofs. The night before the duel, he sent to his friend Auguste Chevalier an account of his work including numerous important theorems without proof. The sixty pages constituting the collected works of Galois appeared, fifteen years after they were written, in the Journal de mathématiques (1846), and in (uvres mathématiques D’ EVARISTE GALOIS, avec une introduction par M. Emile Picard, Paris 1897. Ad ALGEBRAIC INTRODUCTION. [Cu. V ber or we may demand a positive number; for constructions by elementary geometry, we may admit square roots, but not higher roots of arbitrary positive numbers. In the study of a given equation, we naturally admit into the investigation all the irra- tionalities appearing in its coefficients; for example, V3 in con- sidering 2?+(2—5V3)e+2=0. We may agree beforehand to admit other irrationalities than those appearing in the coefficients. In a given problem, we are concerned with certain constants or variables (2) R’, R”,..., Rw together with all quantities derived from them by a finite number of additions, subtractions, multiplications, and divisions (the divisor not being zero). The resulting system of quantities is called the domain of rationality * (R’, R”’,..., R™). | Exampte 1. The totality of rational numbers forms a domain. It is con- tained in every domain Ff. For if w be any element ~0 of R, then w+w=1 belongs to R; but from 1 may be derived all integers by addition and sub- traction, and from these all fractions by division. Exampte 2. The numbers a+bi, where i= —1, while a and b take all rational values, form a domain (7). But the numbers a+bi, where a and b take only integral values do not form a domain. DEFINITION. An equation whose coefficients are expressible as rational functions with integral coefficients of the quantities R’, R”’,..., R will be said to be algebraically solvable (or solvable by radicals) with respect to their domain, if its roots can be de- rived from R’, R”,... by addition, subtraction, multiplication, division, and extraction of at root of any index, the operations being applied a finite number of times. 51. The term rational function is used in Galois’ theory only * Rationalitdtsbereich (Kronecker), Korper (Weber), Field (Moore). + If we admitted the extraction of all the pth roots, we would admit the knowledge of all the pth roots of unity. This need not be admitted in Galois’ theory (see § 89, Corollary). Src. 52] THEORY OF ALGEBRAIC EQUATIONS. 45 in connection with a domain of rationality R. An integral rational function for F of certain quantities wu, v, w, ...1S an expression (3) Die Gap EU ai, , i, i k, ae, \ where 7,7, k,... are positive integers, and each coefficient Cy,... is a quantity belonging to R. The quotient of two such functions (3) is a rational function for R. Thus, 3u+ 2 is a rational function of u in (2), but not in (1). 52. Equality. As remarked in § 49, two expressions involving only constants are regarded as equal when their numerical values are the same. Consider two rational functions plu, v; UW, ys oe Pu, v; W, 8 3 with coefficients in a domain R=(R’, R’”,..., R®). In case R’, R’’,... are all constants, we say that ¢ and ¢ are equal if, for every set of numerical values u,, v,, w,,... which u, v, w,... can assume, the resulting numerical values of ¢ and ¢ are equal. In case R’, R”’,..., R” depend upon certain independent variables r,r’,..., 7™, we say that ¢ and ¢ are equal if, for every set of numerical values which u,v, w,..., 7, 7”,..., 7™ may assume, the resulting numerical values of ¢ and ¢ are equal. When not equal in this sense, ¢ and ¢ are said to be distinct or different. For example, if w and v are the roots of x?+2or+1=0, the functions u+v and —2puv are rational functions in the domain (p), and these rational functions are equal. DerFIniTion. A rational function ¢(2%,,..., %) is said to be unaltered by a substitution s on %,..., x, if the function Pp(%,---, Tn) is equal to @ in the sense just explained. For brevity, we shall often say that ¢ then remains numerically un- altered by s. If 2,, 2,..., 2 are independent variables, as in Lagrange’s theory, and if ¢, is identically equal to ¢, 1.., for all values of 2,,..., 2m, we say that ¢ remains formally unaltered by s. For examples, see § 49. 58. The preceding definitions are generalizations of those employed in the Lagrange theory. The so-called general equation 46 ALGEBRAIC INTRODUCTION. . [Cu. V of degree n may be viewed as an extreme case of the equations (1) whose coefficients ¢,,..., C, are rational functions in the domain (R’, R’,..., R™). In fact, since its coefficients are independent variables belonging to the domain, they may be taken to replace an equal number of the quantities R’, R’”,... defining the do- main, so that the general equation appears in the form an Reeth Re 2+ 2. + RO, Its roots are likewise independent variables (p. 101), so that two rational functions of the roots are equal only when identically equal. 54. Reducibility and irreducibility. An integral rational func- tion F(x) whose coefficients belong to a domain R is said to be reducible in & if it can be decomposed into integral rational factors of lower degree whose coefficients likewise belong to #; irreducible in F if no such decomposition is possible.* ExampteE 1. The function 2?+1 is reducible in the domain (7) since it has the factors x+7 and x+17, rational in (7). But 2?+1, which is a rational function of x in the domain of rational numbers, is irreducible in that domain. cee ir x*+1 is reducible in any domain to which either nv 4 2, or V 7 or 2, Or cae belongs, but is irreducible in all other domains. In fact, its linear factors are c+e, r+ie=x+e?; while every quadratic factor is of the form x?+1, or z*+ar+1, a =+2. If (x) is reducible in R, F(x)=0 is said to be a reducible equation in #; if f(x) is irreducible in R, F(x)=0 is said to be an irreducible equation in R. 55. THroreM. Let the equations F(x)=0 and G(x)=0 have their coefficients in a domain R and let F(x) =0 be irreducible in R. If one root of F'(x)=0 satisfies G(x) =0, then every root of F(x)=0 satisfies G(x)=0 and F(x) vs a divisor of G(x) in R. After dividing out the coefficients of the highest power of z, let ee (x—¢,)(a—&,) . ety G(x) =(@#— 1m)... (@— Hm). * A eEod to decompose a given integral function by a finite number of rational operations has been given by Kronecker, Werke, vol. 2, p. 256. Src. 55] THEORY OF ALGEBRAIC EQUATIONS. 47 At least one ¢ equals an 7. Let €,=7,,..., &.=7,, while the remaining ¢’s differ from each 7. Then the function B(ry=(e—f,)... (x—€,) =(2—7,) .. . (42 — 7) is the highest common factor of F(x) and G(«#). But Euclid’s process for finding this highest common factor involves only the operation division, so that the coefficients of A(x) are rational functions of those of F(a) and G(x) and consequently belong to the domain R. Hence F(x%)=A(x)-Q(x), where A(x) and Q(x) are integral functions with coefficients in R. Since F(x) is irreducible in FR, Q(x) must be a constant, evidently 1. Hence F(«)=H(z), so that F(x) is a divisor of G(x) in R. Corotuary I. If G(x) is of degree en—t, then G(xz)=0. A root of an irreducible equation in # does not satisfy an equation of lower degree in Rf. CorouuarRy II. If also G(x)=0 is irreducible, then G(x) is a divisor of I(x), as well as I’(x) a divisor of G(x). If two irreducible equations in FR have one root in common, they are rdentical. CHAPTER VI. THE GROUP OF AN EQUATION, EXISTENCE OF AN 7!-VALUED FUNCTION; GALOIS’ RESOLVENT. 56. Let there be given a domain # and an equation (1) f(x) =a" —c,0"—1+6,2"-7— ... +(—1)"en=0, whose coefficients belong to R. We assume that its roots 4%, Yo,..+, %m are all distinct.* It is then possible to construct a rational function V, of the roots with coefficients in R such that V, takes n! distinct values under the m! substitutions on 2, ,..., Xn. Such a function is V,=M,2,+MH+ ... +MnIn, if m,,...,M%, are properly chosen in the domain R. Indeed, the two values V, and Vj», derived from V, by two distinct sub- stitutions a and 0b respectively, are not equal for all values of M.,.++,Mn, Since 2,,...,%, are all distinct. Siygise eee possible to choose values of m,,...,m™, in R which satisfy none of the 4n!(n!—1) relations of the form V,=Vj,,. Then from an equation Vy,-=V, will follow a’ =a. As an example, consider the equation x*+2?+a+1=0, with the roots x=—1, t= +1=+/—I, X3= —1, and let R be the domain of all rational numbers. The six functions ‘ 4 —™M,+1M,—1M3, —M,—1M,+1Mz, 1M,—M,—MM,, , —1Mm,+1m,—M,,° —im,—mM,+1mM;, 1m,—iIm,—M., * Equal roots of F(x) =0 satisfy also F'’(x) =0, whose coefficients likewise belong to R, and consequently also H(x) =0, where H(z) is the highest com- mon factor of F(«) and F’(x). If F(x) +H (x) =Q(a), the equation Q(x) =0 has its coefficients in R and has distinct roots. After solving Q(x) =0, the © roots of F(x) =0 are all known. 48 Src. 57] THEORY OF ALGEBRAIC EQUATIONS. 49 arising from the 3! permutations of x,, x,, x3, will all be distinct if no one of the following relations holds: M,—m,=0, m—m,=0, m,—m,=0, (i+1)m,—2im,+(C—1)m, =0, (@—1)m,+(0¢+1)m,—2im, =0, . (t—1)m,—21m,+(¢+1)m, =0, (1+1)m,+(—1)m,—2im, =0, —2im,+ (t—1)m,+ (1+ 1)m, =0, —2im, +(¢7+1)m,+ (i—1)m,=0, of which the last six differ only by permutations of m,, m,, m;. We may, for example, take m,;=0 and any rational values ~0 for m, and m, such that m,~cm,, where c is 1, +7, 1 +7,4(1+7). Thus mz,4+2, is a six-valued function in FR if m, is any rational number different from 0 and 1. [In the domain (7), we may take m,x,+2,, where m,~0, 1, +2, 141, 2(147).] 57. The n! values of the function V, are the roots of an equation (4) PN IeatV SV iV —¥)... Va Va, whose coefficients are integral rational functions of m,,..., Mn, C;, +++, €n With integral coefficients and hence belong to the domain R (§ 50). If F(V) is reducible in R, let F,(V) be that irreducible factor for which /’,(V,)=0; if F(V) is irreducible in R, let F,(V) be F(YV) itself. Then (5) FY(V)=0 is an wrreducible equation called the Galois resolvent of equation (1). Recurring to the example of the preceding section, take Ve=7,—4)," Vo=—2,—13, - Ve=%3— 7}. Then the six values of V, are +V,, +V., +V;, where Watt l, V¥,=21, Vyao~itl. The equaticn (4) now becomes eet CHV ?—V,7)(V?2—V 5°) =(V?—2)(V? +4)(V? +22) =V°+4V4+4V?+16=0. The irreducible factors of F(V) in the domain of rational numbers are Weta =(V—V,\(V+V,), V2-2V+2=(V—Vi(V -F,;), V242V4+2=(V+V,(V+V,). The Galois resolvent (5) is therefore FAV) =V2—2V +2=0. [For the domain (7), the Galois resolvent is V-—V,=V—i—1=0.] 50 THE GROUP OF AN EQUATION. [Cu. VI 58. THrorem. Any rational function, with coefficients im a domain R, of the roots of the given equation (1) ts a rational function, with coefficients in R, of an n!-valued function V,: (6) | P(X, Tq). +» Ln) = O(V,). Let first the coefficients c,,...,¢, In equation (1) be arbitrary quantities so that the roots 2,,...,2, are independent variables. We may then apply the proof in §$ 31 of Lagrange’s Theorem, taking for ¢ the function V, which is unaltered by the identical substitution alone, and obtain a relation (6”) i A el: where F’(V) is the derivative of /’(V) defined by (4). We next give to c,,..., Cy, their special values in #, so that z,, . . ., %, become the roots of the given equation. Since F’(V,)40, relation (6’) becomes the desired relation (6), expressing ¢ as a rational function of V, with coefficients in R. CoROLLARY. I} s be any substitution on the letters x,,..., Xn, then (7) Ps(Xy, Lo, .--, tn) =O(V 5), provided no reduction* in the form of ®(V,) has béen made by means of the equation I'(V,)=0 of § 57. As an example, we recur to the equation 23+2?+2+1=0, and seek an expression for the function ¢=2, in terms of V,=2,—a,. Then F(V) =V°+4V!44V?+16, F’(V)=6V5+16V?+8V, ) 7 =K v5 § a) vy - ks v3 v3 vy VY—FW) 1 Voy, V+V, 1 V—V, VV, ee = —2V°—4V4—12V3—8V?—16V —48, upon setting 7,= —1, x,=1, 7,=—1, V;=71+1, V,=21, V;=—7i+1. Hence _ AV) _ 2 5-4 12) See eFC 5 6V,5+16V,°+8V, In verification, we find that AV ,) =AQti+1) = —481—-16, F’(V,) =161—48, O(V,) =1=2,. Ly = 0(V;,). * That such a reduction invalidates the result is illustrated in the example | of § 59. Src. 59] THEORY OF ALGEBRAIC EQUATIONS. 51 In view of the corollary, we should have %,=0(—V,), 1=9(V,), 2,=9(V;), 2%,=9(—V,), 2,=9(—YV,). To verify these results, we note that 16.—48 _ N80! _ =80 ro ge a Bor cae while O(V;) and O(V,), 9(—V;), and 9(—V;,), xz, and 2, are conjugate imaginaries, and 2, is real. o-V)= 59. As a special case of the preceding theorem, the roots of the given equation are rational functions of V, with coefficients in R: (8) 1,=0,(V,), t2=¢2(V1),..-, In=Gn(Vj). Hence the determination of V, is equivalent to the solution of the given equation. Since each V, is a rational function of 2,, .. ., , with coefficients in R, it follows that all the roots of the Galois resolvent are rational functions with coefficients in R of any one root V,. Exampte. For the equation x*+2*+x+1=0, and V,=2,—2, we have 2,=—1; 2,=V,—1, %3=—V,+1, V,=2V,-2, V,=—V,+2. Although x, and V,—1 arenumerically equal, the functions z,and —V,—1, obtained by applying the substitution (z,2,), are not equal. The relation 2,=V,—1 is a reduced form of z,=@(V,), obtained in virtue of the identity V ?—2V,+2=0 ($57). Thus —2V,5—4V,*—12V,’—8V,?—16V, —48 a) —48V, +32, 6V P+ 167° +8V, a 16V, — 64, —48V,+32_(—3V,+2)(Vi+2)_—3V—4V,+4__—10V,+10 16V,—64 a (V,—4)(V,+2) V’—2V,-8 —10 =V,-1. It happens, however, that the equality 7,=V,—1 leads to an equality x,=V,;—1=—V,+1 upon applying the substitution (a,7;). The fact that the identical substitution and (2,7;), but no other substitutions on 2, 2, 23, lead to an equality when applied to z,=V,—1 finds its explanation in the general theorems next established. 52 THE GROUP OF AN EQUATION. (Cu. VI THE GROUP OF AN EQUATION. 60. Let the roots of Galois’ resolvent (5) be designated (9) ve Va; a7 Foe ey Var the substitutions by which they are derived from V, being (10) |e AY ey get h These substitutions form a group G, called the group of the given equation (1) with respect to the domain of rationality R. The proof consists in showing that, if r and s are any two of the substitutions (10), the product rs occurs among those substi- tutions. Let therefore V, and V, be roots of (5). Then eV le 0: Now VJ, is a rational function of V, with coefficients in R: (11) V,=YV,), the function 0 being left in its unreduced form as determined in § 58. Hence F'{0(V,)]=0, so that one root V, of the equation (5) irre- ducible in R satisfies the equation (12) {AV )]=9, with coefficients in R. Hence (§ 55) the root V, of (5) satisfies (12). io. OE) | =0- In view of the corollary of § 58, it follows from (11) that (V,)s=Vrs=A(Vs). Hence [’,(V;s)=0, so that V,, occurs among the roots (9). ExamptLeE, For the equation v*+2?+2-+1=0 and the domain R of rational numbers, the Galois resolvent was shown in § 57 to be V?—2V +2=0, having the roots V; and V;._ Since V; was derived from V, by the substitution (2,2), the group of the equation x*+2?+2+1=0 with respect to R is {J, (apa,)}. For the domain (7), the Galois resolvent was shown to be V—V,=0. Hence the group of the equation with respect to (7) is the identity. Src. 61] THEORY OF ALGEBRAIC EQUATIONS. 53 61. The group G of order N of the equation (1) with the roots 2%, X,.. +, Xn possesses the following two fundamental properties: A. Every rational junction (2, %,..., tm) of the roots which remains unaltered by all the substitutions of G lies in the domain R. B. Every rational function $(4,, %,...,%n) of the roots which equals a quantity in R remains unaltered by all the substitutions of G. By a rational function 6=¢(a,,..., 2») of the roots is meant a rational function with coefficients in R. Then by § 58 (18) P=O(V,), Pa=P(Va), Po=P(Vs),.--, P= O(V2), where @ is a rational function with coefficients in R. Prooj of A. If d6=¢a=dp= .. . = Gj, it follows from (18) that b= TOV) FOV) OM) +... +00}. The second member is a symmetric function of the N roots (9) of Galois’ resolvent (5) and hence is a rational function of its coeffi- cients which belong to R. Hence ¢ lies in R. Proof of B. If ¢ equals a quantity r lying in R, we have, in view of (138), the equality 0(V,)—r=0. Hence J, is a root of the equation, with coefficients in &, (14) 0(V)—r=0. Since one root V, of the irreducible Galois resolvent equation (5) satisfies (14), all the roots V,, Va,..., Vz of (5) satisfy (14), in view of § 55. Hence fe j—-7—0, O(V,)—r=0, ..., OVp)—r=0. It therefore follows from (13) that 6=¢d.=q¢)=...=¢i. Hence ¢ remains unaltered by all the substitutions of G. 62. By arational relation between the roots 2, ..., % 1s meant an equality f(2,,...,%n)=¢(%,,..-,%n) between two rational functions, with coefficients in R. Then d6—¢is a rational function, 54 THE GROUP OF AN EQUATION. [Cu. VI equal to the quantity zero belonging to R, and therefore (by B) is unaltered by every substitution s of G. Hence d,—¢,=$—¢=0, so that d6,=¢,. Hence the result: Any rational relation between the roots remains true if both members be operated upon by any substitution of the group G. ExampLe. For the domain of rational numbers, it was shown in § 60 that the equation z°+2?+2+1=0 has the group {J, (a,,)}. The rational relation (§ 59, Example) leads to a true relation 2,=x,—x,—1=V,—1 under the substitution (425). If we apply (2,2,), we obtain a false relation xz, =2,—z,—1. 63. THrorEM. Properties A and B completely define the group G of the equation: any group having these properties vs rdentical with G. Suppose first that we know of a group G’={I, a’, ,..., m} that every rational function of the roots 2,,..., 2%, which remains unaltered by all the substitutions of G’, liesin R. The equation F(V)=(V—V)(V—Va)(V—Vy)..- (V—-Va)=0 has its coefficients in R since they are symmetric functions of V,, Va’,.--, Vm and therefore unaltered by the substitutions of G’. Since /’’(V)=0 admits the root V, of the irreducible Galois resolvent (5), it admits all the roots V,, Va,..., V, of (5). Hence I, a,..., l occur among the substitutions of G’, so that @ is a subgroup of G’. | Suppose next that we know of a group GAT We UG ag that every rational function of 2,,..., X, which lies in R-remains unaltered by all the substitutions of G’’. Then the rational function F(V,), being equal to the quantity zero lying in R, remains un- siteredsby-a’ , 0571s... wie. a0 Lua 0=F (VV) =F (Va)=P (Vor = ... =F Vir) e Src. 64] THEORY OF ALGEBRAIC EQUATIONS. > 55 Hence V,;Va”,..., V-’ occur among the roots V,, Va,..., V; of F(V)=0. Hence G” is a subgroup of G. If both properties hold for a group, G’=G”; then G’ contains G as a subgroup and G’ is a subgroup of G. Hence G’=G”=G. It follows that the group of a given equation for a given domain as unique. In particular, the group of an equation is independent of the special n!-valued function V, chosen. EXAMPLE. For the equation x?+x?+x+1=0 and the domain F of all rational numbers, the functions +V,, +V., +V; of y 57 are each 6-valued Employing V,, we obtain the Galois resolvent (V—V,)(V —V;) =V?—2V +2=0 and the group {/, 2,2,)}. Evidently no change results from the employment of V;. If we employ either —V, or —V;, we obtain the Galois resolvent (V+Vi)(V+V3) =V?+2V+2=0 and the group {J, (x,7,)}. If we employ either V, or —V., we get (V—V.)(V+V,) =V?+4=0. Since V,=2z,—2;, the substitution replacing V, by —V, is (a3), so that the group is again {J, (2,.,)}. ACTUAL DETERMINATION OF THE GROUP G OF A GIVEN EQUATION. 64. Group of the general equation of degree n. Its coefficients C1). ++, are independent variables, and likewise its roots (p. 101). We proceed to show that, jor a domain R containing the coefficients and any assigned constants, the group of the general equation of degree n 1s the symmetric group Gp. It is only necessary to show that the Galois resolvent /’,(V)=0 is of degree n!. In the relation F(V,)=0, we replace V, and the coefficients ¢,..... ¢, by their expressions in terms of x,,...,2,. Since the latter are independent, the resulting relation must be an identity (see p. 101) and hence - remain true after any permutation of 2,,...,%. By suitable permutations, V, 1s changed into V,,.... V»; in turn, while ¢,,..., ¢n, being symmetric functions, remain unaltered. Hence /,(V,)=0, ., F(Vn1)=0. Hence F,(V)=0 has n! distinct roots. Another proof follows from § 63 by noting that properties A and B hold for the symmetric group G,,; when 2,,..., %» are inde- 56 THE GROUP OF AN EQUATION. [Cu. VI pendent variables. Thus A states that every symmetric function of the roots is rationally expressible in terms of the coefficients. 65. To determine the group of a special equation, we usually resort to some device. It is generally impracticable to construct an n!-valued function and then determine the Galois resolvent (5) ; or to apply properties A and B directly, since they relate to an infinite number of rational functions of the roots. Practical use may, however, be made of the following lemma, involving a knowledge of a single rational function: Lemma. J} a rational function (a,,...,%pn) remains formally unaltered by the substitutions of a group G’ and by no other substi- - tutions, and if ¢ equals a quantity lying in the domain R, and 7 the conjugates of & under Gy, are all distinct, then the group of the given equation for the domain R is a subgroup of G’. In view of the first part of § 68, it is only necessary to show that every rational function $(2,,..., 2%), which remains numeri- cally unaltered by all the substitutions of G’, ies in R. If G’ is of order P, we can set b=btb.t ... +4P); so that ¢ can be given a form such that it is formally unaltered by all the substitutions of G’. Then, by Lagrange’s Theorem (§ 31), ¢@ is a rational function of ¢ and hence equals a quantity lying in fF. EXAMPLE 1. To find the group of z?—1=0 for the domain R of all rational numbers. The roots are a=1, %=3(—1++/-8), ,=4(—1—*/—3). Taking ¢ =2,, it follows from the lemma that Gis a subgroup of G’ = {J, (a2,)}. Since x, does not lie in R, Gis not the identity (property A). Hence G=@’. EXAMPLE 2. To find the group G of y3—7y+7=0 for the domain R of all rational numbers. For the cubic y?+py+q=0, we have (§ 2) D=(Y1—Y2)"(Y2—Ys) "(Ys Yi)? = — 279? —4p* For p= —7, g=7, we get D=7*. Hence the function $= (Y1—Y2) Y2—Ya) Ys —Yy) Src. 65 THEORY OF ALGEBRAIC EQUATIONS. 57 has a value +7 lying in #& and its conjugates ¢ and —¢ under G;, are distinct. By the lemma, G is therefore a subgroup of the alternating group G;, and hence either G, itself or the identity G,. Now, if the group of the equation were G,, its roots would lie in R. But * a rational root of an equation of the form y?—7y +7 =0, having integral coefficients and unity as the coefficient of the highest power, is necessarily an integer. By trial, +1, +7 are not roots. Hence the roots areall irrational. Hence the group G is G. EXAMPLE 3. Find the group of z'+1=0 for the domain of rational numbers. We seek a rational function of the roots x, 7, 73, x, which equals a rational number. Let us try the function y,=2%,2,+2%,2, Specializing the result — holding for the general quartic equation (§ 4), we find that, for the quartic x*+1=0, the resolvent equation (16) for y, is y>—4y=0. By a suitable choice of notation to distinguish the roots x:, we may set Yi=—2, Y2=0, Ys=+4+2. Hence y, equals a rational number and its conjugates under G,, are all distinct. Hence G is a subgroup of G;, the group to which 2,2,+2,7, belongs formally (§ 21). Similarly, by considering the conjugate functions y,=2,7,+2,%,, and y;=2,%,+2,2,, we find that G is a subgroup of Gj and Gj’. Hence G is a subgroup of G,(§ 21). Hence G is G,, G,, — Gy =[L, (& 4X2) (X32) }, Gh [T, (2123) (2%) }, or GE = [T, (ayy) (as) }. Now G#G,, since no root of «!+1=0 is rational. If G,, consider ¢,=2,+%,—2,—2, For the general quartic equation x*+ax>+bx?+cx+d=0, we have t,?=a’—4b+4y, by §5. Hence, for z'+1=0, t,?=—8. Since ¢, is not rational, G#G. If G;’, consider t,=2,+2,—2,—2;. In general, t,2=a?—4b+4y,;. Here t,2,=+8. Since é, is not rational, G#G,’. » If G), consider ,=2,+2,;—2,—2, In general, é,2>=a?—4b+4y,. Here t,2=0. Since a conjugate —é, of t, equals ¢,, no conclusion may be drawn from the use of ¢,. But ¢=2,7,;—2,2, is unaltered by Gj. Now df? = (2103+ 2,0,)’ —427,0,0,0,=Y,” -—4 = —4, Hence ¢ is not rational, so that GG}. The group of z*+1=0 for the domain of rational numbers is therefore G. EXERCISES. Find for the domain of rational numbers the group of 1, z?+2?+2+1=0 (using the lemma, § 65). 2. (x—1)(4+1)(x—2) =0. * Dickson, College Algebra (John Wiley & Sons), p. 198. 58 THE GROUP OF AN EQUATION. [Cu. VI 3. 23—2=0. [2,, X, v7, and (1, —2,)(x,—23)(%3—2,) are irrational. ] 4, e4+a3+2?+2+1=0 with roots 7,=¢, %,=¢«7, 7%,=e*, 7,=6*, where ¢ is an imaginary fifth root of unity. Since the resolvent for 2,27,+ a2, is y®—y?—3y+2=0 with the roots 2, $(—1+/5), G is a subgroup of G%. The latter has the subgroup C,= {J, (1234), (13)(24), (1432) }, to which belongs fy =2 12%, 42,20, +2,°u,+2,72, Here d,=e'+ e8+¢e+e?=—1 is rational. The six conjugates to ¢, under G,, are distinct; they are obtained from ¢, by applying I, (12)(34), (12), (14), (23), (34); their values are —1, 4, 1+2¢e+?, 14+2e8+e4 142e?+c¢, 1424+? respectively. Hence G is a subgroup of C,. To G,={J, (13)(24)} belongs (x,—2x, +22, —ix,)? =(1+2i)(e? + 68 — et —e) = 4.4/5 (1421). Hence G#¥G}. Evidently G#G,. Hence G=C,. 5. Show that, for the domain (1, 7), the group of #!+1=0 is G4. 6. Show that, for the domain (1, w), w=imaginary cube root of unity, the group of z?’—2=0 is C,;={I, (x,12%3), (2427) }. Hint: (2, +w2,+wx,)? and (7,+w?x,+w2,)? have distinct rational values. TRANSITIVITY OF GROUP; IRREDUCIBILITY OF EQUATION. 66. A group of substitutions on n letters is transitive if it contains a substitution which replaces an arbitrarily given letter by another arbitrarily given letter; otherwise the group is intran- sitive. Thus the group G,= {I, (272) (a3%,), (24%) (Xo%%,), (2%,)(%_%3)} is transitive ; I replaces x, by 2,, (x,%_)(x,%,) replaces x, by 22, (1,%3)(%,%,) replaces x, by Zz, (x,x,)(x,%,) replaces x, by x, Having a substitution s which replaces 2, by any given letter z; and a substitution ¢ which replaces x, by any given letter 2;, the group necessarily contains a substitution which replaces 2; by 2;, namely, the product s—1. The group H,={J, (x,%), (x3%), (,%,)(%,%,)} is intransitive. 67. THEOREM. The order of a transitive group on n letters is divisible by n. Of the substitutions of the given group G, those leaving 2, unaltered form a subgroup H={I, h,,..., h,}. Consider a rect- angular array (§ 28) of the substitutions of G with those of H in the first row, choosing as g, any substitution replacing x, by a, as g, any substitution replacing x, by x,, etc. Then all the sub- stitutions of the second row and no others will replace a, by 2,, > a ok aso = ne en a gw gr pe coe ee af ———— ain LR Src. 68] THEORY OF ALGEBRAIC EQUATIONS. 89 all of the third row and no others will replace xz, by x, ete. Since G is transitive, there are v=n rows. But the order of G is divisible by v (§ 26). Examples of transitive groups: G;@), G,@), G9, G4, G@, GO. The least order of a transitive group on 7 letters is therefore n. A transitive group on 7 letters of order 7 is called a regular Broup. Thus G,\® and G,( are regular. 68. THrorEM. I] an equation is irreducible for the domain R, its group for R ws transitive; if reducible, the group is intransitive. First, if f(a) =0 is irreducible in R, its group for £# is transitive. For, if intransitive, G contains substitutions replacing 2, by 4%, T>,.+-, tm, but not by rm4,,..-, Ln, the notation for the roots being properly chosen. Hence every substitution of G permutes Z14,-++, Um amongst themselves and therefore leaves unaltered any symmetric function of them. Hence the function g(x)= (w—2,)(t—2X)...(*—X») has its coefficients in R, so that g(x) isa rational factor of f(x), contrary to the irreducibility of f(z). Let next /(x) be reducible in & and let g(x) =(a—2,) .. . (x—Xm) be a rational factor of j/(x), m being is a rational function of d with coefficients in R. The function ¢ belongs to a certain subgroup H of G, say of index v. By means of a rectangular array of the substitutions of G with those of H in the first row, we obtain the v distinct con- jugate functions ¢,,¢,,..., ¢, and a set of functions ¢,, d,..., dy, not necessarily distinct, but such that a substitution of G which 62 THE GROUP OF AN EQUATION. [Cu. VI replaces ¢; by ¢; will replace ¢; by ¢; (compare § 31). If g(d) be defined by (15), then =a) (P+ e+ ae +e) is an integral function of t which remains unaltered by all the substitutions of G, so that its coefficients lie in & (§ 71). Taking ¢,=¢ for t, we get P=A(Y) +9). For examples, see § 58. The function V, is unaltered by the identical substitution only, which leaves unaltered any rational function. REDUCTION OF THE GROUP BY ADJUNCTION. 78. For the domain R=(1) of all rational numbers, the group of the equation 2°+2?+24+1=0 is G,={J, (x,73)}; while its group for the domain R’ =(1, 7) is the identity G, (see § 60). In the language of Galois and Kronecker, we derive the domain R’ =(1, 2) from the included domain R =(1) by adjoining the quantity 7 to the domain Rf. By this adjunction the group G, of «3+2?+2+1 is reduced to the subgroup G,. The adjoined quantity 7 is here a rational function of the roots, 7=x7,=—vzs, in the notation of § 49 for the roots. The Galois resolvent V?—2V +2=0 for R becomes reducible in R’, viz., (V—i—1)(V +7—1) =0. For the domain R=(1), the group of x*+1=0 is G,; for the domain (1, 2), its group is the subgroup Gj={J, (a,x3)(a,27,)}, by § 65. By the adjunc- tion of 7 to the domain R, the group is reduced to a subgroup Gj. Here (=2,? =2,? = —2x,?= —2,?=2,7,, in the notation of §49. The subgroup of G, to which «,? belongs is Gj. If we afterwards adjoin +/2, the roots will all belong to the enlarged domain (1, 7, \/2), so that the group reduces to the identity. For example, 7,=(1+7) SND: For the domain R=(1), the group of z?—2=0 is G,; for the domain (1, w), w being an imaginary cube root of unity, the group is the cyclic group C, (Exercises 3 and 6, page 58). Call the roots 4,=8/2, 2%, =w8/2=02,, 2%, =w*/2=wx,. Then w=2,/z,, a rational function belonging to C;. In fact, (x,x.%3) replaces X,/t, by X/%,=W=7,/%X,, (2X37) replaces x,/z, by 2,/%;—G =a these two substitutions and the identity are the only substitutions leaving z,/t, unaltered. If we subsequently adjoin ¥/2, the roots all belong to the enlarged domain (1, w, ¢/2), so that the group reduces to the identity. Src, 74] THEORY OF ALGEBRAIC EQUATIONS. 63 74. In general, we are given a domain R=(R’, R”,...) and an equation /(z)=0 with coefficients in that domain. Let G be its group for R. Adjoin a quantity €. The irreducible Galois resolvent F’,(V)=0 for the initial domain R may become reducible in the enlarged domain R,=(¢; R’, R”’,...). Let a(V, §&) be that factor of F,(V) which is rational and irreducible in R, and vanishes for V=V,. If V,, Va,..., V,are the roots of A(V, €)=0, then G’= {I,a,...,k} is the group of f(x) =0 in R, (§ 57). Hence G’ is a subgroup of G, including the possibility G’=G, which occurs if F,(V) remains irreducible after the adjunction of &, so that AV, €)=F(V). THEOREM. By an adjunction, the group G is reduced to a sub- group G’. 75. Suppose that, as in the examples in § 73, the quantity adjoined to the given domain RF is a rational function ¢(x,, 2%, ...,%n) of the roots with coefficients in R. TuHrorEeM. By the adjunction of a rational function $(a,,...,%n) belonging to a subgroup H of G, the group G of the equation is reduced precisely to the subgroup H. It is to be shown that the group H has the two characteristic properties (§ 61) of the group of the equation for the new domain ieee ,...). Hirst, any rational function (2), ...,2n) which remains unaltered by all the substitutions of H is a rational function of ¢ with coefficients in R (§ 72) and hence lies in Ay. Second, any rational function f(a, ..., YZ») which equals a quantity p in &, remains unaltered by all the substitutions of H. For the relation ¢=o may be expressed as a rational relation in & and hence leads to a true relation when operated upon by any sub- stitution of G (§ 62) and, in particular, by the substitutions of the subgroup H. The latter leave ¢, and hence also p, unaltered. Hence the left member ¢ of the relation remains unaltered rela all the substitutions of H. CHAPTER VII. SOLUTION BY MEANS OF RESOLVENT EQUATIONS, 76. Before developing the theory further, it is desirable to obtain a preview of the applications to be made to the solution of any given equation /(x)=0. Suppose that we are able to solve the resolvent equation (15), one of whose roots is the rational function ¢ belonging to the subgroup H of the group G of f(x) =0. Since ¢ is then known, it may be adjoined to the given domain of rationality (R’, R’”,...). For the enlarged domain Aj= (J; R’, R”,...), the group of j(7)=0 is H. _ Let yan be a rational function with coefficients in R, which belongs to a subgroup K of H. Suppose that we are able to solve the resolvent equation one of whose roots is y. Then y may be adjoined to the domain R,. For the enlarged domain R,=(y, ¢; R’, R”’,...), the group of j(z)=0 is K. Proceeding in this way, we reach a final domain FR; for which the group of /(x)=0 is the identity G,. Then the roots 2,,...,%n, being unaltered by the identity, lie in this domain R; (property A, §61). The solution of /(x)=0 may there- fore be accomplished if all the resolvent equations can be solved. To apply Galois’ methods to the solution of each resolvent, the first step is to find its group for the corresponding domain of rationality. 77. Isomorphism. Let G be the group of a given equation f(x)=0 for a given domain Rk. Let ¢(x,,...,%,) be a rational function of its roots with coefficients in RF and let ¢ belong to a subgroup H of index v under G. Consider a rectangular array 04 Src. 77) THEORY OF ALGEBRAIC EQUATIONS. 65 of the substitutions of G with those of H in the first row, and the resulting functions conjugate to ¢: hy=L ch, EP d= 92 hog, «+ hpg, f= “Po, qv (xt ae : hpg, e. Apply any substitution g of the group G to the v conjugates (16) $1 Par Pog +++s Pore The resulting functions (17) $9, Yo,9) Poqus age? E Yo,0 are merely a permutation of the functions (16), as shown in § 29, hence to any substitution g of the group G on the letters x,,..., 2p, there corresponds one definite substitution r=(5 ae Ho) (‘2 ) $y Yong =e Yo,9 Yoi9 on the letters (16). We therefore obtain * a set J’ of substitutions y, not all of which are distinct in certain cases (xs. 2 and 3 below). TurorEM. The set I’ of substitutions 7 jorms a group. For to g, g’, and gg’ correspond respectively r= ($2 Hie: = (5% 4} 7 = (Ses 3) Yoo)” Yaa Yajao' To compute the product 77’, we vary the order cf the letters in the first line of 7’ and have r-(be,) =i) 2 Hence if J" contains y and 7’, it contains the product 77’. Since J” contains a substitution replacing ¢ by ¢,, for any t=1,..., v, the group I’ is transitive (§ 66). * For a definition of J’ without using the function ¢, see § 104. 66 SOLUTION BY MEANS OF RESOLVENT EQUATIONS. [Cu. VII | DEFINITIONS. The group J’is said to be isomorphic to G, since to every substitution g of G corresponds one substitution 7 of I’, and to the product gg’ of any two substitutions of G corresponds the product yy’ of the two corresponding substitutions of J”. If, inversely, to every substitution of J" corresponds but one substi- tution of G, the groups are said to be simply isomorphic;* other- wise, multiply isomorphic.* EXAMPLE 1. Let G=G,@), H=G,, $=2,+w2,+wx,. Set (compare § 9) hi=$, tr=¢a, $:=$o, fu=Pe, Ys=d, Po=Pe- Then a=(2,%,2;) replaces ¢, by ¢, =w¢,, and ¢, by ¢,=w¢,. Hence areplaces gy by wf, =¢5, $3 by w$,=¢1, $e by wh,=95, $s by w7f,=¢,. Hence to a corresponds a=(¢,¢.¢3)(GaPe;). Similarly, we find that to c=(2#,%3) corre- sponds 7 =(¢,44)(¢2¢5)(Y¢3¢¢). Hence to b=a? corresponds 6 =a’, to d=a-'ca vorresponds 0 =a—'ya, to e=b~'cb corresponds ¢«=/—17. We have therefore the following holoedric isomorphism between G and I’: if i G = (X,X2X3) a=(Pibos) (Yahos) b= (242322) B=(Pidshe) Yupse) C= (X2X3) r= (Piha) Yobs) (Pao) d = (2,23) 0 = (Poe) (YsPs) (Pres) € = (X42) € = ($345) Pipe) Yoh) It may be verified directly that to b, d, e correspond ~, 6, ¢, respectively. Since I, a, 8, 7, 0, e replace ¢, by ¢,, ¢2, ¢3, Yu, Ys, Ye, respectively, J is tran- sitive. . EXAMPLE 2. Let G=G,,(, H=G,, ¢=(2,—2,)(a,—2,). Set Pi=P, $2=(%1—X3)(%—X), Y3=(%,—X,) (12 —T3). We obtain the following meriedric isomorphism between G and I’: I, (2X) (g%q), (yy) (TeX), (Hq) (T_T) | I (x2030,), (24230), (11423), (2,222) (Yifoys) (2224X3), (2,242), (X,1275), (1324) (Yifs¢2) The group J’ is transitive since it contains substitutions replacing ¢, by fi; ho, or ?s. * Other terms are holoedric and meriedric for simple and multiple — isomorphism. Src. 78} THEORY OF ALGEBRAIC EQUATIONS. 67 78. Order of the group J’. To find the number of distinct substitutions in J’, we seek the conditions under which two sub- stitutions 7 and 7’ of /' are identical. Using the notation of § 77, the conditions are Yo,0=Vo;0 (i=1, 2, en, v), if we set g,=I. Applying to this identity the substitution g-1g;~, we get i $= g,9'9—19,—1- Hence g,g’9~'g;-'=h, where h is some substitution leaving ¢ unal- tered and hence in the group H. Then / gg *=9;_ "hg; Cae eer ae) But g;—*hg; belongs to the group H;=g,~'Hg; of the function ¢,; (§ 39). Hence g’g~' belongs simultaneously to H,, H,,..., Hy, and therefore to their greatest common subgroup J. Inversely, any substitution o of J leaves ¢,, ¢.,...,¢, unal- tered and hence corresponds to the identity in J’. Then g and g’ =og correspond to substitutions 7 and 7’ which are identical. If G is of order k and i the greatest common subgroup J of H,, He, , 1, 1s of order 7, then I’ 1s of order k/j. EXAMPLE 1. For G=G,, H =G,, the order of I is 6 (§ 77, Ex. 1). Exampte 2. For G=G‘), H=G, (§ 77, Ex. 2), we have H,=H,=H,, since G, is self-conjugate under G,, (§ 41). Hence k=12, 7=4, so that the order of J’ is 3. EXAMPLE 3. For G=@Q), H, =G,, $ =2,4,+7,%,, we set (§ 29, Ex. 2) Py =U 11, +%3%y, Po=XyX3tXyX, y= 2XyX_tXyXp. hen f,—G,, H,—G,, H,=—G,, J=G, (§ 21). Hence I is of order 24=6. This result may be verified directly. There are only 6 possible substitutions on 3 letters ¢,, %, ¢;. But the substitutions of G which lead to the identical substitution of J’ must leave ¢,, ¢,, ¢, all unaltered and hence belong to the greatest common subgroups G, of H,, H,, H;. Hence exactly four substitu- tions of G correspond to each substitution of J",so that the order of I’ is 24 =6, The four substitutions of any set form one row of the rectangular array for 68 SOLUTION BY MEANS OF RESOLVENT EQUATIONS. [Cu. VIL G,, with the substitutions I, (x,%2)(%3%4), (aX 3) (a%,), (184) (as) of G, in the first row. As right-hand Pe we a take G1=1, g2=(L2%324), = (20423), = (324), J5=(X2%q), Jo =(L2%s). To the Foe substitutions of the first a the four of the second row,..., correspond I, (diheds), (Yidsh2), (Pods), (dis), (Yi). 79. Of special importance is the case in which H,, H,,..., Hy are identical, so that H 7s self-conjugate under G. Then J=H, so that the order k/7 of I’ equals the index »v of H under G. Hence the number of distinct substitutions of J” equals the number of letters ¢,,...,¢. upon which its substitutions operate, or the order and the degree of the group I’ are equal. Moreover, I" was seen to be transitive. Hence J" isa regular group (§ 67). DEFINITION.* When H is self-conjugate under G, the group I’ is called the quotient-group of G by H and designated G/H. In particular, the order of G/H is the quotient of the order of G by that of H. EXAMPLE 1. By Examples 1 and 2 of § 77, the quotient-group G,/G, is a regular group on six letters; the quotient-group G,,/G, is the cycle group {L, (Gidrs), (YiPs¢2)}, which is a regular group. EXAMPLE 2. We may not employ the symbol G,,/G;, since G, is not self-conjugate under G,, (§ 78, Ex. 3). EXAMPLE 3. Consider the groups G, and G, on three letters. To G, belongs ¢, =(a,—2,)(@,—23)(«;—2,); under G, it takes a second value ¢, = —¢, ($9). We obtain the following isomorphism between G, and I’: if (X15), (X,%32) | L (X25), (X23), (X4X2), ($2) Since G; is self-conjugate under Gs, we have I'=G,/G,={I, (¢,¢,)}. CoroLtuary. J] H is a selj-conjugate subgroup of G of prime index v, then I’ 1s a cyclic group of order v (§ 27). Illustrations are afforded by the groups G,,/G, and G,/G, of Exs. 1 and 2. Remark. Any substitution group G is simply isomorphic with a regular group. In proof, we have merely to take as ¢ any n!- valued function V,, whence I’ will be of order equal to the order of G. * Holder, Math, Ann., vol. 24, page 31. Src. 80] THEORY OF ALGEBRAIC EQUATIONS. 69 80. Let H be a maximal self-conjugate subgroup of G (§ 48). The quotient-group ['=G/H is then simple (§ 43). For if I’ has a self-conjugate subgroup 4 distinct from both J’ and the identity G,, there would exist, in view of the correspondence between G and I’, a self-conjugate subgroup D of G, such that D contains H but is distinct from both G and H. This would contradict the hypothesis that H was maximal. For example, if H is a self-conjugate subgroup of G of prime index yr, it is necessarily maximal. Then I’ is a cyclic group of prime order v (Cor., § 79) and consequently a simple group. 81. The importance of the preceding investigation of the group I of substitutions on the letters ¢,, ¢., . . .,¢, lies in the significance of I’ in the study of the resolvent equation (15) g(y)=(y—P)Y—$2) -- - (Y— Gv) =9, whose coefficients belong to the given domain R. We proceed to prove the THroreM. lor the domain R, the growp of the equation (15) is I’. We show that J’ has the characteristic properties A and B of §61. Any rational function p(¢,, ¢,...,¢.) with coefficients in R may be expressed as a rational function r(x, %;...,2%n) with coefficients in R: (18) oY) Qo; cy yj =1(2y, Voy eee) In). From this rational relation we obtain a true relation (§$ 62) upon applying any substitution g of the group G on a,,...,%. But g gives rise to a substitution 7 of the group J'on ¢,,...,¢,. Hence the resulting relation is (19) Or(Pry Poy + + « : Py) =P (2, La). +) En). To prove A, let o(¢,,...,¢.) remain unaltered by all the sub- stitutions of I’, so that o,=, for any y in I’. Then, by (18) and (19), 7,=7, for any ginG. Hence r lies in the domain RF (property A for the group G).' Hence a lies in R. To prove B, let po lie in the domain R. Then, by (18), r lies 70 SOLUTION BY MEANS OF RESOLVENT EQUATIONS. [Cu. VII in R. Hence rg=r, for any g in G (property B for the group G). Hence, by (18) and (19), o,=, so that o remains unaltered by all the substitutions 7 of I’. Cor. 1. Since J’ is transitive ($ 77), equation (15) is irreducible in R (§ 68). This was shown otherwise in § 71. Cor. 2. If the group H to which ¢ belongs is self-conjugate under G, the group of the resolvent (15) is regular (§ 79). The resolvent is then said to be a regular equation. Cor. 3. If H is a self-conjugate subgroup of G of prime index », the group of (15) is cyclic (§ 79, Corollary). The resolvent is then said to be a cyclic equation of prime degree »v. Cor. 4. If H is a maximal self-conjugate subgroup of G, the group of (15) is simple (§ 80). The resolvent is then said to be a regular and simple equation. 82. THEOREM. The solution of any given equation can be reduced to the solution of a chain of simple regular equations. Let Gbe the group of the given equation for a given domain R, and let a series of composition (§ 48) of G be GH eae as the factors of composition being A (index of H under G), » (index of K under H),...,o Gndex of G, under M). Let @)0 ewe be rational functions of the roots belonging to H, K,...,M, G,, respectively (§ 70). Then ¢ is a root of a resolvent equation of degree A with coefficients in &, which is a simple regular equation (§ 81, Cor.4). By the adjunction of ¢ to the domain R, the group G of the equation is reduced to H (§ 75). Then ¢ is a root of a simple regular equation of degree » with coefficients in the enlarged domain (¢, #). By the adjunction of ¢, the group is reduced to K. When, in this way, the group has reduced to the identity G,, the roots z,,...,2%p lie in the final domain reached (compare § 76). In particular, if the factors of composition i, u,...,0 are all prime numbers, the resolvent equations are all regular cyclic equations of prime degrees (§ 81, Cor. 3). Src. 83] THEORY OF ALGEBRAIC EQUATIONS. 71 83. THEOREM. A cyclic equation of prime degree p ts solvable by radicals. Let & be a given domain to which belong the coefficients of the given equation /(z)=0 with the roots 2,2%,,...,2p -,, and for which the group cf /(2) =O is the cyclic group G= {/,s,s?,...,s?—1}, where s=(%,2,%,...2»-,). Adjoin to the domain # an imaginary pth root of unity * w and let the group of /(z)=0 for the enlarged domain R’ he G’. Consider the rational functions, with coefficients ic.1t;, (20) G,= 2) +02, +0%2,+ ... tw Me, _,. Under the substitution s, #; is changed into w~6;. Hence 67 =6; is unaltered by s and therefore by every substitution of G and of the subgroup G’ (§ 74). Hence 9; lies in the domain Ki’ (§ 61). Extracting the pth root, we have 0;=~/0,. Since the function (20) belongs to tke identity group, it must be possible, by Lagrange’s Theorem (§ 72), to express the roots 2%,2,,...,%p-, rationally in terms of J;. The actual expressions for the roots were found in the following elegant way by Lagrange. We have, by (20), Ly tXy +2, Hictas ert yess C = wary Qytwr, +72, ave Piha eye NG, 9 ah i hey Gytwe, tots, +... +w%P-32,_,—V90, é 2 ho to twtr, tw De, +... tw? ay = VOp-1 where c= 6, is the negative of the coefficient of xP—! in f(x) =0. Multiplying these equations by 1, w™, w~™,..., w~—*, respect- ively, and adding the resulting equations, and then dividing by p, we get T n= * } ctw V6, +0-#Y/6,+ an + w—P-MN/ Oy 3 l , ? ) * As shown in § 89, w can be determined by a finite number of applications of the operation extraction of a single root of a known quantity. Poucel toot... +ae—)t=—0 fori=1,2,...,p—1. 72 SOLUTION BY MEANS OF RESOLVENT EQUATIONS, [Cu. VII for i=0, 1,..., p—1. The value of one of these p—1 radicals, say / 6,, may be chosen arbitrarily; but the others are then fully determined, being rationally expressible in terms of that one. Indeed, | V6; -(VO,)'=0;+6 becomes w*0;+(w—10,)* upon applying the substitution s and hence is unaltered by s, and is therefore in the domain R’. 84. From the results of $$ 82-83, we have the following THEOREM. I} the group of an equation has a series of composition for which the factors of composition are all prime numbers, the equation is solvable by radicals, that 1s, by the extraction of roots of known quantities. The group property thus obtained as a sufficient condition for the algebraic solvability of a given equation will be shown (§ 92) to be also a necessary condition. CHAPTER VIII. REGULAR CYCLIC EQUATIONS; ABELIAN EQUATIONS. 85. Let f(a) =0 be an equation whose group G for a domain R consists of the powers of a circular substitution s=(a,x7,... 2p): CYR a epics e n being any integer. Since the cyclic group G is transitive and of order equal to its degree, it is regular (§ 67). Inversely, the gen- erator s of a transitive cyclic group is necessarily a circular sub- stitution on the n letters.* The equation /(z)=0 then has the properties: (a) It is irreducible, since its group is transitive (§ 68). (b) All the roots are rational functions, with coefficients in R, of any one root 2,. Indeed, there are only n substitutions in the transitive group on 7 letters, and consequently a single substitution (the identity) leaving x, unaltered. Since 2, belongs to the identity group, the result follows by Lagrange’s Theorem (§ 72). Let xz,=O(x,). To this rational relation we may apply all the substi- tutions of G (§ 62). Hence (21) %=O(%,), Tz=O(%q), -.-, Ln=O(Xn_1), X= O(In). DEFINITION. An irreducible equation for a domain R between whose n roots exist relations of the form (21), @ being a rational function with coefficients in FR, is called an Abelian equation.t * A non-circular substitution, as t=(2x,%,%,)(7,%;), generates an intransi- tive group. Thus the powers of ¢ replace x, by 2, x, or x; only. t+ More explicitly, uniserial Abelian (einfache Abel’sche, Ironecker). A more general type of ‘‘ Abelian equations” was studied by Abel, Gfuvres, I, No. XI, pp. 114-140. 73 74 REGULAR CYCLIC EQUATIONS ; ABELIAN EQUATIONS. [Cu. VIII 86. THrorEM. The group G of an Abelian equation is a regular cyclic group. Denote any substitution of the group G by —) [(tpte te Coe eee ee Tq Ug Uy ... BM)" Applying to the rational relations (21) the substitutions g (§ 62), %,=O(Xa); t;=O(x,), ; > 2. =0G eee But, by (21), 0(a2)=2a4,, holding also for a=n if we agree to set U;=Lian=Tign= .-. Lt follows that La= UT a+; XL =Xotay «sens Ta=Ly44. Since the equation is irreducible, its roots are all distinct. Hence, aside from multiples of n, B=atl1, p=Pt+1l=a+2, d=7+1=a43,... ae é Sy o5 oT een ) -g Vig Lat, Bare! wi alelnn ae Since g replaces x; by 2;144_,, it is the power a—1 of the circular substitution s=(2,%1%3..¥%n) which replaces x; by 2;,,. Hence G is a subgroup of G’={/, s, s?,..., s”-*}. But Gis transitive, since the equation is irreducible. Hence G=G’. Jal EXAMPLE. The equation 7*+2?+2?+2+4+1 = =0 has the roots Yy=&, =e, wz=e*, m=, where ¢ is an imaginary fifth root of unity. Hence T.=%y?, %3=X,?, Ly=X;”, 1 =2,?. Moreover, the equation is irreducible in the domain F of all rational numbers (§ 88). This may be verified directly by observing that the linear factors are «—e? and hence irrational, while ett+a3+o?4+e4+1=(2?+axr+r)(a?+b241-') gives at+b=1, ab+r+r—'=1, ar—'+br=1, so that either a=4(14V5), b=}1FV5), r=, r 1 pend be r+1, (ea a Hence the group for F is a cyclic group. Compare Ex. 4, page 58. or, a rtrtrt+r+ti1 =o. Src. 87) THEORY OF ALGEBRAIC EQUATIONS. 75 87. Cyclotomic equation for the pth roots of ibis p being prime, (22) ePTitaP ... +2e+1=0. Let ¢ be one root of (22), so that «?=1, «Al. Then (23) Smee | See el * are all roots of (22) and are all distinct. Hence they furnish all the roots of (22). As shown in the Theory of Numbers, there exists,* for every prime number p, an integer g such that g”—1is divisible by p for m=p—1 but not for a smaller positive integer m. Such an integer g is called a primitive root of p. It follows that the series of integers SS eicnsoreed pemmer when divided by p, yield in some order the remainders Pee 2 an ay 1 Hence the roots (23) may be written 2 2— Y= &, L= 89, Le= 9,11, Up_y HEF P 0. The last coefficients in the products are 105 ie Crs + w2,"1, er ee Xn, = + w2x,"2, respectively, since the roots of 7?—A=0 are (26) Eas GIL Ge, Wa ee w being an imaginary pth root of unity. But the last coefficients, and their quotient +w°%x,”, where m=n,—n,>0,liein R. Since p and mare relatively prime, integers and v exist for which mu—pv=1. *. (w°t,™) P= ta Pe = tA a = Ae where x’ is one of the roots (26). Hence A,z2’, and consequently x’, liesin R. Then A equals the pth power of a quantity z’ in R, contrary to assumption. Hence 2?—A must be irreducible. 91. THEOREM. A binomial equation of prime degree p, zP—A=0, can be solved by means of a chain of Abelian equations of prime degree. Let R be the given domain to which A belongs. Adjoin w and denote by R’ the enlarged domain. Then the roots (26) satisfy the relations of the type (21) of $85, A(x) being here the rational function wa. The discussion in § 90 shows that x?—A is either irreducible in the enlarged domain f’ or else has all its roots in R’. In the former case, the group of x?—A=O for RF’ is a regular cyclic group (§ 86); in the latter case, the group for R’ is the identity. But w itself is determined by an Abelian equation (§ 87). Hence, in either case, xz? —A=0 is made to depend upon a chain of Abelian equations, whose degrees may be supposed to be prime (§ 89). CHAPTER IX. CRITERION FOR ALGEBRAIC SOLVABILITY. 92. We are now in a position to complete the theory of the algebraic solution of an arbitrarily given equation of degree n, (1) 2y—u! A group property expressing a sufficient condition for the algebraic solvability of (1) was established in § 84. To show that this property expresses a necessary condition, we begin with a dis- cussion of equation (1) under the hypothesis that it is solvable by radicals, namely (§ 50), that its roots 2,,...,2, can be derived from the initially given quantities R’, R’,... by addition, sub- traction, multiplication, division, and extraction of a root of any index. These indices may evidently be assumed to be prime numbers. If €, 7,..., ¢ denote all the radicals which enter the expressions for all the roots 2, 2,,..., %», the solution may be exhibited by a chain of binomial equations of prime degree: Poh giv, ..), ni=M(e, RR’, ...), «oes ge= Pi. oe) I g, Lae Calle a oF x,= RY, cee 7) o FR’, R”, a) (i=1, -.+,%), L, M,..., P, R; being rational functions with integral coefficients, in which some of the arguments €, 7,... written may be wanting. By $91, each of these binomial equations, and therefore also the complete chain, can be replaced by a chain of Abelian equations of prime degrees: 79 80 CRITERION FOR ALGEBRAIC SOLVABILITY. (Cu. IX Diy hei owe = Os Abelian for domain R; WES Od re R”,. . =O, Abelian for (y, ay : Abie - ne: ‘bie R’,. eas Abelian fomame we 4 BY: a= Od ee Roe (ta Bt. We begin by solving the first Abelian equation O(y)=0 and adjoining one of its roots, say y, to the original domain R; the group G of (1) then reduces to a certain subgroup, say H, including the possibility H=G (§ 74). Then we solve the second Abelian equation Y(z)=0 and adjoin one of its roots, say z, to the enlarged domain (y, #); the group H reduces to a certain sub- group, say J, including the possibility J=H. Proceeding in this way, until the last equation 0(w)=0 has been solved and one of its roots, say w, has been adjoined, we finally reach the domain (w,..., 2, y, R), with respect to which the group of (1) is the identity G,, since all the roots x; lie in that domain. By every one of these successive adjunctions, either the group of equation (1) is not reduced at all or else the group is reduced to a self-conjugate subgroup of prime index. This theorem, due to Galois, is established as a corollary in the next section; its impor- tance is better appreciated if we remark that each adjoined quantity is not supposed to be a rational function of the roots, in contrast with § 75, so that we shall be able to draw an important conclusion, due to Abel, concerning the nature of the irrationalities occurring in the expressions for the roots of a solvable equation ($ 94). From this theorem of Galois, it follows that the different groups through which we pass in the process of successive adjunction of a root of each Abelian equation in the chain to which the given solvable equation was reduced must form a series of composition of the group G of the given equation having only prime numbers as factors of composition. Indeed, the series of groups beginning with G and ending with the identity G, are such that each is a self- conjugate subgroup of prime index under the preceding. Hence the sufficient condition (§ 84) for the algebraic solvability of a Src. 93] THEORY OF ALGEBRAIC EQUATIONS. 81 given equation is also a necessary condition, so that we obtain Galois’ criterion for algebraic solvability: In order that an equation be solvable by radicals, it is necessary and sufficient that its group have a series of composition in which the factors of composition are all prime numbers. 93. Theorem of Jordan,* as amplified and proved hy Holder: + For a given domain R let the group G, of an equation F(x) =0 be reduced to G,’ by the adjunction oj all the roots of a second equation F(x)=0, and let the group G, of the second equation be reduced to G./ by the adjunction of all the roots of the first equation I’,(x)=0. Then G,{ and G,! are self-conjugate subgroups of G, and G, re- spectively, and the quotient-groups G,/G,' and G,/G,/ are simply asomor phic. Let ¢,(&, €,..-, €n) be a rational function, with coefficients in R, of the roots of the first equation which belongs to the sub- group G,’ of the group G;, of the first equation (§ 70). By hypothe- sis, the adjunction of the roots 7,, 7.,...,%m of the equation F(x) =0 reduces the group G, to G,’.. Hence ¢, lies in the enlarged domain, so that (27) PSs, Sar - +25 End=PilMy Nay -- +> Nm)s the coefficients of the rational function ¢, being in R. Let ¢,, ¢.,..., ¢%~ denote all the numerically distinct values which ¢, can take under the substitutions (on €,,..., &n) of G. Then G,’ is of index k under G, ($71). Let ¢,, d,..., 6; denote ‘all the numerically distinct values which ¢, can take under the substitutions (on 7,,...,%m) of G,. The k quantities ¢ are the roots of an irreducible equation in R (§ 71); likewise for the / quantities ¢. Since these two irreducible equations have a com- mon root ¢, = ¢,, they are identical (§ 55, Cor. II). Hence ¢,,..., dx coincide in some order with ¢,,..., $7; in particular, k=l. If s; is a substitution of G, which replaces ¢, by its conjugate ¢,, then s; transforms G,’, the group of ¢, by definition, into the group of ¢; of the same order as G,’. But ¢;, being equal to a ¢d, lies in * Traité des substitutions, pp. 269, 270. + Math. Annalen, vol. 34. 82 CRITERION FOR ALGEBRAIC SOLVABILITY. [Cu. IX the domain R’=(R; 7,, ..., Nm), and hence is unaltered by the substitutions of the group G,/ of the equation /’,(~#)=0 for that domain R’ (§ 61, property B). Hence the group of ¢; contains all the substitutions of G,’; being of the same order, the group of ¢; 1s identical with G,’. Hence G,’ is selj-conjugate under G,. The group of the irreducible equation satisfied by ¢, is therefore the quotient-group G,/G,’ (§ 79). Let H, be the subgroup of G, to which belongs ¢,(,, qo, . . - ; Nm)- Since ¢, is a root of an irreducible equation in F of degree 1=k, the group H, is of index k under G, (§ 71). By the adjunction of ¢d, (or, what amounts to the same thing in view of (27), by the adjunction of ¢,), the group G, of equation /’,(~) =0 for R is reduced to H, (§ 75). If not merely ¢,(€,,...,-&,), but all the €’s them- selves be adjoined, the group G, reduces perhaps further to a subgroup of H,. Hence G,’ is contained in H,. We thus have the preliminary result: If the group of F',(2)=0 reduces to a subgroup of index k on adjoining all the roots of F',(7)=0, then the group of F,(x)=0 reduces to a subgroup of index k,, k,=k, on adjoining all the roots of F(x) =0. Interchanging /’, and F’, in the preceding statement we obtain the result: If the group of /’,(z)=0 reduces to a subgroup of index k, on adjoining all the roots of /’,(2)=0, then the group of F,(x)=0 reduces to a subgroup of index k,, k,>k,, on adjoining all the roots of F,(%)=0. Since the hypothesis for the second statement is identical with the conclusion for the first statement, it follows that k,=k, k, Sk, ky ky, so that k,=k. Hence the group G,/ of the theorem is identical with the group H, of all the substitutions in G, which leave ¢, unaltered. It follows that G,’ 7s self-conjugate under G, (for the same reason that G,’ is self-conjugate under G',). The irreducible equation in Ff satisfied by ¢, has for its group the quotient-group G,/G,!. But the two irreducible equations for FR satisfied by ¢, and ¥,, respectively, were shown to be identical. Hence the groups SEc. 94] THEORY OF ALGEBRAIC EQUATIONS. 83 G,/G, and G./G,/ differ only in the notations employed for the letters on which they operate, and hence are simply isomorphic. CorouuARY. For the particular case in which the second equa- tion is an Abelian equation of prime degree p, all of its roots are rational functions in # of any one root, so that by adjoining one we adjoin all its roots. By the adjunction of any one root of an Abelian equation of prime degree p, the group of the given equation I’ (a)=0 either is not reduced at all or else is reduced to a self- conjugate subgroup of index p. 94. If G, is simple and if the adjunction causes a reduction, then G, is reduced to the identity. Hence the group G,’=H),, to which belongs ¢,, is the identity. Hence the roots 7,, 7, ..-, Mm of /’,(2)=0 are rational functions in FR of ¢, (§ 72) and therefore, in view of (27), of the roots £,,..., €n of F,(x)=0. Ij the group of an equation F',(x)=0 for a domain R is reduced by the adjunction of all the roots of an equation F’,(x) =0 whose group for R is simple, then all the roots of F',(x)=0 are rational functions im KR of the-roots of F(x) =0. Since the group of a solvable equation j(#)=0 has a series of composition in which the factors of composition are all prime num- bers, the equation can be replaced by a chain of resolvent equations cach an Abelian equation of prime degree (end of § 82, § 85). The adjunction of a root of each resolvent reduces the group of the equation and the group of the resolvent is simple, being cyclic of prime order. Hence the roots of each Abelian resolvent equation are all rational functions of the roots of f(z)=0. But the radicals entering the solution of an Abelian equation of prime degree are rationally expressible in terms of its roots and an imaginary pth root of unity (§ 83), 4/0, = 2) tue, +w2,+ Te le ge ae er a and hence are rationally expressible in terms of the roots of f(x) =0 and pth roots of unity. We therefore state Abel’s Theorem: The solution of an algebraically solvable equation can always be performed by a chain of binomial equations of prime degrees whose 84 CRITERION FOR ALGEBRAIC SOLVABILITY. (Cu. IX roots are rationally expressible in terms of the roots of the given equation and of certain roots of unity. The roots of an algebraically solvable equation can therefore be given a form such that all the radicals entering them are rationally expressible in terms of the roots of the equation and of certain roots of unity. This result was first shown empirically by Lagrange for the general quadratic, cubic, and quartic equations (see Chapter I). The Theorem of Abel supplies the step needed to complete the proof of the impossibility of the algebraic solution of the general equation of degree n >4 (§ 48). 95. By way of illustrating Galois’ theory, we proceed to give algebraic solutions of the general equations of the third and fourth degrees by chains of Abelian equations. For the cubic x?—c,x?+¢,7—c,=0, let the domain of rationality be R=(c,, C,, ¢,). The group of the cubic for R is the symmetric group G, (§ 64). To the subgroup G, belongs A = (1 —X_)(L_—X3)(X3—2,). In view of Ex. 3, page 4, 4 is a root of the equation (28) A? =c,7c,” + 18¢,¢,C, —4¢,3 — 4c,3c, — 27,7. Its second root —Z is rationally expressible in terms of the first root 4, and (28) is irreducible since 4 is not in R& for general ¢,, ¢, cs. Hence (28) is Abelian ($85). By adjoining 4 to R, the group reduces to G, (§ 75). Solve the Abelian equation w?+w+1=0 (§ 87) and adjoin w to the domain (4, R). To the enlarged domain R’=(w, 4, ¢,, CG, ¢3) belong the coefficients of the function J, =2,+wx,+ wr. By § 34, ¢,? has a value lying in Rf’, namely, $F =4[2c,3 —9c,c, + 27c, —3(w—w?) A]. This binomial is an Abelian equation for the domain R’ (§ 91). By the adjunction of ¢,, the group of the cubic reduces to the Sxc. 96] THEORY OF ALGEBRAIC EQUATIONS. 85 igentbys Hence z,, 2, 7, lie in the domain (¢,, w,'4, ¢,, c, ¢,). Thus, by § 34, oo oe n=t(atht" di, "), Xo =3(at0% to ae). 1 We may, however, solve the cubic without adjoining w. In the domain (4, ¢,, ¢, ¢,), the cubic itself is an Abelian equation, since its group G, is cyclic (§ 85). By the adjunction of a root 2, of this Abelian equation, the group reduces to the identity, so that %, and x, must lie in the domain (2,, 4, ¢,, ¢&, ¢,). The explicit expressions for x, and 2, are given by Serret, Algébre supérieure, lew. NO. O11: Lo 3 { (6c, —2c,”) x," + (9e,—7¢,C.+2¢,°— 4) x, + 4¢,”—¢,7c,—3¢,C,+¢,4}, the value of x, being obtained by changing the sign of 4 throughout. 96. For the general quartic x*+a2°+ bx?+cx+d=0, the group for the domain R=(a, b, c, d) is G.,. To the subgroup G,, belongs = (X, —Xp)(1, — Lg) (Ly — Ly) (Lp — L3) (Lo — Lq) (Lg — 4). Since 4? is an integral function of a, b, c, d with rational coefficients (§ 42), we obtain 4 by solving an equation which is Abelian for R. After the adjunction of 4, the group is G,. To the subgroup G, cf G,. belongs the function y,=2,7,+2,2,. It satisfies the cubic resolvent equation (§ 4) (16) y>— by? + (ac—4d)y—a*d+4bd—c?=0. The group of this resolvent for the domain (4, a, b, c, d) is a cyclic group of order 3 (§ 79, Cor.), so that the resolvent is Abelian. By the adjunction of y,, the group of the quartic reduces to G,. To the subgroup G, of G, belongs the function t=a,+2,—2,—a, It is determined by the Abelian equation (§ 5) (29) Me i= 0745 £47), By the adjunction of t, the group reduces to G,. To the identity subgroup G, of G, belongs 2,; it is a root of (17), § 4: w+ 4(a—t)x+ hy, — (say, —0)/t=0. 86 CRITERION FOR ALGEBRAIC SOLVABILITY. (Cu. 1X After the adjunction of a root x, of this Abelian equation, the eroup is the identityG,. Hence (§ 72) all the roots lie in the domain (x,,t, y,,4,a,6,c,d). This is evident for x,, since 2,+2,= —4(a—2). For z, and x,, we have Vet Lg=Lj+X,—t, %—2y=(Y2— Ys) + (2, —M), while y, and yz are rationally expressible in terms of y,, 4, and the coefficients of (16), as shown at the end of § 95. In fact, (Y1—Y2)(Yo—Ys)(Yi— Ys) has the value 4 by § 7. 97. Another method of solving the general quartic was given in § 42. For the domain R= (a, a, b, c, d), where w is an imaginary cube root of unity, the group is G,, (§ 64). After the adjunction of 4, the group is G.. To the self-conjugate subgroup G, belongs f=Y, + wy,+wy;, where y,=2,7,+27,2,, etc., so that ¢, is a rational function of 2,, 2, 73, %,, with coefficients in R. By § 42, $= 3(w—w?) A — 2160, so that ¢, 1s determined by an equation which is Abelian for the domain (4, w, a, b, c, d). Then, by § 42, y;, y., ys; belong to the enlarged domain (¢,, 4, w, a, b, ¢, d). By the adjunction of t, a root of the binomial Abelian equa- tion (29), the group reduces to G,. By the adjunction * of both t=V —1 and V=a2,—2,+i2,—ix,, which is a root of a binomial quadratic equation (§ 42), the group reduces to the identity G,. The expressions for x,, 2%, 2%, %, In terms of #; V, 1, and a, are given by formula (41), in connection with (40), of § 37. * Without adjoining 7 and V, we may determine t,=2,+27,—x,—2, from t,2=a?—4b+4y,. Then t,=2,+2,—x,—2; is known, since t,t,t;=4ab —8c—a? by formula (39) of § 36, where t,=¢. Then x,=1(—a+i,+i,4+4,), %,=}(—a+t,—t,—t3), etc. CHAPTER X. METACYCLIC EQUATIONS; GALOISIAN EQUATIONS. 98. Analytic representation of substitutions. Given any sub- stitution ga (To TM 22 +++ ny De aly ele ie ie so that a,b,..., k form a permutation of 0,1,..., n—1, it is pos sible to construct a function (z) of one variable z such that fO)=a, P(1)=b, (2=c, ..., s(m—1)=k. Indeed, such a function is given by Lagrange’s Interpolation- Formula, ak (z) bF(z) a kF(2) #2) = 2870) DED) + eee + (2—n4+ 1)’ (n—1)? where F'(z)=2(z—1)(2—2)...(g—n+1) and F’(z) is the deriva- tive of /’(z). Then the substitution s is represented analytically We confine our attention to the case in which n is a prime num- ber p, and agree to take 7,=%,4p=%z4.p= .... » Then (as in § 86) the circular substitution t=(a, 7, 2,...2%p_,) May be represented in the form t ( z ) ° Ve+1 Let G be the largest group of substitutions on a, 2,..., Lips 87 88 METACYCLIC AND GALOISIAN EQUATIONS. (Cu. X under which the cyclic eroup H={I,t,t?,...,i?—*} is self-conjugate. The general substitutions g of G and h of H may be written x HY ee ie )=e J Ss Vz+a By hypothesis, g~'tg belongs to H and hence is of the form 2’. ~1_ ($02) ee) 1 = (740) ih : é ), d Tz 41)’ ft! Lp(z+1) But t” replaces xyz) by %@)4a- Hence must Lp(z +1) = T(z) +a Taking in turn z=0, 1, 2,..., and writing 6(0) =), we get L(y) =Vb+a, VUd(2)= (1) +a=Vb+2a, Vo(3)= Vb(2)+a = Vh+3ay By simple induction, we get 42)=2%p+za for any integer z. Hence ly 2D oe) . Here a and }=¢(0) are integers. Also a is not divisible by 7, since g~tg is not the identity. The distinct substitutions * g are obtained by taking the values a=1,2,..5 ,p—1;-0=0, 12. ee The resulting p(p—1) substitutions form a group called the meta- cyclic group of degree p. This follows from its origin or from (ia) (eecsg) 7 eee) er Laz+b Laz4+ 3 La(az+b)+ B Xaaz+(ab+ f) , Remark. The only circular substitutions of period p in the metacyclic group are the powers of t. For a=1, (80) becomes ?#; for a1, (30) leaves one root unaltered, namely, that one whose — index z makes az+b and z differ by a multiple of p. * Formula (30) does, indeed, define a substitution on 2X, 2, ..., @n—:, (7 Ly ots Danke a ) XT VXatb Xoat+b..- Z since b, a+b, 2a+b,..., (p—1)a+b give the remainders 0, 1, 2, .., p—l, in some order, when divided by p. In proof, the remainders are all different. Src. 99] THEORY OF ALGEBRAIC EQUATIONS. 89 99. A metacyclic equation of degree p is one whose group G for a domain Ff is the metacyclic group of degree p. It is irre- ducible since G is transitive, its cyclic subgroup H being transitive. Again, all its roots are rational functions of two of the roots with coefficients in #&. For, by the adjunction of two roots, say zu and z,, the group reduces to the identity. Indeed, if g leaves tu and x, unaltered, then (au+b)—u, (av+b)—v are multiples of p, so that their difference (a—1)(uw—v) is a multi- ple of p, whence a=1, and therefore b=0. Hence the identity alone leaves xu and 2y unaltered. DerinitTion. Fora domain &, an irreducible equation of prime degree whose roots are all rational functions of two of the roots is called a Galoisian equation. Hence a metacyclic equation 1s a Galovsian equation. 100. Given, inversely, a Galoisian equation of prime degree 7, we can readily determine its group G for a domain R. The equa- tion being irreducible, its group is transitive, so that the order of G is divisible by p ($ 67). Hence G contains a cyclic subgroup H of order p (see foot-note to § 27). Let 2) and x, denote the two roots in terms of which all the roots are supposed to be rationally expressible. Among the powers of any circular substitution of period p, there is one which replaces x, by x,. Hence, by a suitable choice of notation for the remaining roots, we may assume that ‘H contains the substitution aed Ho gt nen eae To show that H is self-conjugate under G, it suffices to prove that any circular substitution, contained in G, r= (x, Xi Li, btu Lin _,) is a power of ¢; for, the transform of ¢ by any substitution of G will then belong to H ($40). Since every two adjacent letters in r are different, ¢,,,—2, 1s never a multiple of p and hence, for at go METACYCLIC AND GALOISIAN EQUATIONS. [CH. X least two values » and »v of z chosen from the series 0, 1,..., p—1 gives the same remainder when divided by p. Hence y] Ue gt Pi, 4 y—tyr BAY, Lk Since r is a power of a circular substitution replacing 2, by 2x,, we may assume that 7,=0, 7,=1. The hypothesis then gives Lj,= 8 al X;,, xi.) (a=0, Ly. eee p=); where @, is a rational function with coefficients in R. Applying to these rational relations the substitutions r“¢ ““ and r’t ™ of the group G, we obtain, by § 62, Diy yin =F alXo, Lk), Vig, ,-i,=FulXq, Le). Hence the subscripts in the left members are equal, so that Vasp—laty=ty—=C (das L, wis De omitting multiples of p. Hence every subscript in r exceeds by c the (u—v)th subscript preceding it. Hence r is a power of ¢. Since G has a self-conjugate cyclic subgroup H, it is contained in the metacyclic group of degree p (§ 98). | The group of a Galovsian equation of prime degree pis a subgroup of the metacyclic group of degree p. 101. A metacyclic equation is readily solved by means of a chain of two Abelian equations. Let $= R(x, 2,,..., %p_,) belong to the subgroup H of G.* Then J=¥¢, J.= R(X, Vo, XL4y vee) Lop 2) a Pp_.= R(X, Up—1) Lop —2) A wi - * 2 7 — £* Ss) ee cee: om | wi ] = 3 01 12 Or ae le vietelete-phes