IMAGE EVALUATION TEST TARGET (MT-3) // Aw/ ^^J* Jts* jib 1.0 I.I 1.25 1^ ill M IIIII2.5 H- 14.0 1.4 2.2 2£ 1.8 1.6 150mm 'W &. / ^> ^a ■5» „V> / /: ^^' O 7 /APPLIED A i IIVMGE . Inc ^ 1653 East Main street =1 Rochester, NY 14609 USA S Phone: 716/482-0300 = Fax: 716/288-5989 / ff*f' Hoots of Pur Uni-Sertal Aheiiau Equations. Bv Gkouo. P.XXOK Yo,.., Unl..;,, C^,,, Toronto, a....,,. e OUJKCT 01. TJJ|.; J'a|.|,,{. «.W.Vwhen ihei-oot, form a ! . , . ''"""""°'' '"">' ""^ '■"»°'' ae„.ee, ,(s roots, m tlio ordinary Abeliai, .lotnlioii, arc a*]. Bxi, d^xi , (j«-i an which the coefficients ,> and , are such that V(- V_ 27.^) is ..f , • pure Abe an, because ■•« ;« „r„ii i ^ ^>" ^'!? ) is rational, is a function of eilher^f «« otherj ' ""^ ™°' "' ""^ '='"^'" '» » '"'--I su.«ct:\f:r™tfr :i'::z:f -'"'- ■?» ^--^«"-« "■= -e.„r,„,K, -■ion Of pnre n„ JJllr^ ^t^l.t'lH -17^ IT " '^'■'- then g ven of the neee^.nw .n.ri «: ■ ''""^"^^^ iS ^^-^ 15). A deduction s ™Hicient fon„s of the roots of the Z^Z^^.a I y """""-^ """ tained by two difl-erent tnethods (? 7-83 T ! f, "'' """'■'"= '"" "''- cient forms of the root, of ,l,„ ■ necessary and sufB- .he continued p od of any „' "'b ""T,-"' ^'"'"" "' " O"^'- "Wch is (§4o-§4«). Tilt ; b : ritd for'ir' ^'■'"'°. -'-■■^ -- '»"^ ^..e wbich is four times the contitrXlrof':; llZtttttl" 220 YouNO : Forms, Necessari/ and Sufficient, of the Roots of pritiies (§47-§57). Finally, from the relation between the solvable irreducible equation of prime degree )i and the pure uni-nerial Abelian ecpiation of degree 71 — 1, the necessary and sufficient forms of the roots of the irreducible solvable equation of prime degree n are shown to be determinable for all cases in which n — 1 is either the continued product of a number of distinct primes, or four times the continued product of a number of distinct odd primes (§58-§64). Preliminary. Corollary from a Lino of Kroncchcr. §4. It was proved by Kronecker that, n being any integer, the primitive w"^ roots of unity are the roots of an irreducible equation, that is, of an irre- ducible equation with rational coefficients. We shall have occasion to make use of the following Corollary from this law : Let w and to' be two primitive n^^ roots of unity, and let F{w) be a rational function of w. Then, if F{7o)=. 0, F{w') = . For, by hypothesis, F{io) = 7iw'-\-hiw'-^ + etc. = 0, whore //, fii, etc., are rational. We assume s to be less than n, and h to be distinct from zero ; therefore h-^F{w)} = IV' + h-^iw'-^ + etc. = 0. Therefore w is a root of the equation ^{x) =z x' + 7r^hix'~^ -^ etc. =0. If i|/ (a) — . be the equation whose roots are the primitive n^^ roots of unity, w is a root of the equation 4^{x) = 0. Therefore the equations 4) (a;) = and'4'(a;) = have a root in common. But, by Kronecker's law, the equation 4- (jc) = is irreducible. Therefore (p{x) is divisible by '^'(a;) without remainder. This implies that all the roots of the equation ■4'(ic) = are roots of the equation ^ {x) = . Therefore ^ (?«') = . Tlierefore F{icf) = . Principles established by Abel, §5. Let/(a;) = be a uni-serial Abelian equation of the w*** degree, and let its roots, in the order in which they circulate, be the terras in (1). It is known (see Serret's Cours d'Algcbre superieure. Vol. II, page 500, third edition) -!- J. i- J. that Xi= a; ■\- Ri -\- r; -ir + -R^-i, Pure Unl-S»:rhd Ahdiun B/uations. )riimtive ?r' tind h to be 227 where E, is a rational function of the primitive .»■ root of nnity ,. an.J of the known quanffes involved in the coefficient, of 0; un.i, . being any integer I IS derived f.-on, R, by channi„g ,, into to'. Putti, ° ' ' intr — — I a; + i= It; -I- w*R^ 4- w^'R; + 4. w"'-i)'ij;;_j ^ /gx the . roots of the equation /(.) = are obtained by giving'. •„ ,,^, ,,,,,,. sively the values 0, 1, 2 „_ i. Therefore nR^ I. the sum of the roots of the equation; consequently, RJ is rational. An equation of the type {RJiT'Y = F{w) .3. subsists for every integral value of ., F{w) bemg a rational function of «, and of the know,! quantities involved in the coelficionts of 0. As .. may be any one of the prun.tive .- roots of unity, if the general primitive n- root of unity )V^f7'. "'y 'TT" " '" ''' '" ^' "^^^"^ ''''' "^ • '^'''« - ^•^"ts of the equa- tion /(..) = will then be obtained by giving f, in the expression 111 '■ r; + io'R; + io"R.;, + etc. (4) successively the values 0, 1, 2 n~-l. Abel's investigation shows that the form of the function F(.) in (3) is independent of the particular primitive n root of unity denoted by .. Hence the change of .. into uf causes equation (3) to become j. ^ {R,,Rrr=F{w'), (5) the symbol i^' having the same meaning for every value of e. ■Fundamental Element of the Ror' . §6. Because R„ R„ etc., are derived from R, by c.anging w into w\ w\ etc the root or, can be constructed when R, is given. We may therefore call i2x he fundamental element of the root. Examples of the way in which the root IS constructed from its fundamental element will present themselves in the course of the paper. A Ca-tam Rational Function of the Primitive n- Root of Unity, n being an Odd ' Prime Number. § 7. Taking n an odd prime number, there is a certain rational function of the primitive «". ^oot of unity to, of which we shall have occasion to make r I i 228 YOUNO: Fuims, Nevcusan/ und SaJjicUul, 0/ titt Jivols of frequent use. It will bo convenient to describe it hero, and to point out two of its properties. Let w, ii:\ w^\ ..... vf'-\ u\) bo a cycle containing all the primitive m'" roots of unity. The number ^ may bo asHumod to be less than n. With a view to convenience iu printing, the indices of the powers of 10 in (G) nuiy bo written 1. >^' «. /^ .'"S, e, 0; (7) that is to siiy, a = X\ fi = ?.^ and so on. Take P^ a rational function of w, and, z being any integer, let J\ bo what /\ becomes when iv is changed into w'. Then the function to which wo desire to call attention is PIP' I*^ P'P'^P (8) The subscripts- of the factors of the expression (8) are the terms in (7), while the indices are the terms in (7) in reverse order. The expression (8) may be denoted by the symbol «^,. From ^„ as expressed in (8). derive ^, by chang- ing 10 into to', z being any integer. Then <^i = PlP-.Pi .... PiP^^P, 0=P{Pi PlP.Pl , (9) The second of these equations is derived from the first by changing lo into w\ This, since a = ^^ and /3 = ;l^ and so on, causes w" to become w\ and to" to become < and so on. Hence it causes P, to become P,, P, to become P^, and so on. Thus the second of equations (9) is obtained. The rest are obtained in a similar manner. §8. One property which the function <^i possesses is that ^f has a rational value. For .'+.... + r-^ = ;n_i — 1 ;,— 1 Because (6) is a cycle of primitive w*" roots of unity, ;i'-i— 1 is a multiple of n. And, since ^ is less than «, ;i - 1 is not a multiple of n; therefore t is a multiple of n. Put t=.mn; then „ . MiKL ;i -iiHamul- *'' = ^V(/V/':..../-). Co.npanng this with the second of eq.mtions (9). ^''creforo ,^.~.s^ Substitute hero the value of A in n n 'Pi . >n (11). Ihen ^^^-' = (P.-"/',-^^)''. Therefore to .. In the .L way we t„ !„ „ i 7 n "''°" '= " "'"'»' """°'- '» ^ - t»e (10) .ub«i.,« for ov',;. iLg::, ::,:„ ,r '"'^'™ '"'" »■■ """"""^ °f «- CriTEUIOK of Pube U»I.SliBIAL AnELIAN.SK. r/ie Crilei-im Slated. §10. A Criterion of nnre uiii-^pri^l aKaIi^ ■ i:: — :t : "- -^-^--^ -^ r.-. f-. =::;: integer, let li, be derived from P K„ ^u • . •'^ .. ,. , , . ^iivea iioiii i,?! by changing *<; into w'. Then if /?" i« rational, and if the terms 7?" /?' «f-> l , " teims iiJ, , i?, , etc., are such that an equation of the type ,mm 230 Yei'Nfi : Fontts, Ni:t;i>>mir)/ nml Sul)kim(, uf tin liuuts of (a) HiibsistH for every iiifegml viiliio of 2, an cruiiitioii (5), in whidi tho Hynibol /''liiis till! hiiiiii! iiit>iinin>; as in (ll), at the Hiiine time subsist in ji; for every value of/ priuii! to It, the tt values of j-,^^ in (2), itlitained by K'vinji >< succijHsiviMy the values (I, 1, 2, n — 1, are the rootn of a pure uni-serial Abeliaii equa- tion, [irovided alwayH that tho e(|uation of the <*"' degree, of which they can bo bIiovvii to be tho roota, is irreducible. Proof of the Criterion, §11. Hero we assume that the conditions Bpeclfied in §10 are satirtfied, and wo have to hIiow that the n values of j-,^, in (2), obtained by putting s succea- sively ecjual to 0. 1, 2, n — 1 , are the roots of a pure uni-serial Abelian equation. § 12. We will lir.st prove that the n values of the expression (4) obtained by civing t siiccossively the n values 0, 1, 2 , n — 1, are the same, the order of the terms not being considered, as the n values of x,^i in (2) obtained by giving a successively the values 0, 1, 2, . . . . , » — 1. Because vf is a primitive n*'' root of unity, all the «*'' root.s of unity distinct from unity are contained in the series W', 10^', w^' , 10 1)1 — 1) « Therefore the two series ill, *^2 I -« ^3 ) • • • • I Jhi — l> liey -"so -"36 1 . . ■ . ) J-i[n-\)tl are identical with one another, the order of tho terras not being considered. Therefore, also, the two series J. ' i J. i. — ' ' ■tie I -l^i'i -^Sel • • • • , Jll/i — Del are identical with one another, the order of the terms not being considered, it being understood that i?j', Ei^) etc., are the same «"' roots of E^, E.^^, etc., or of El, E2, etc., that are taken in the series ^1", i?^ , etc. Let the expression (4) be called Xt+i- The separate members of the expression x, + i are E;, WEI, to^'E;, etc. (13) Takine: ,s- with a definite value, let es = hn -\- c. "1 I ;ho Hynibol tvcry vuliio !.ssiv(My till) sliiiii t'(iua- hey can be tisfiufl, uiid g a Hucccs- ial Abelian btained by I, the order btained by ity distinct considered. risidered, it , etc., or of 3sion (4) be (13) Piin Uni-Saiu/ AUliun J'Jfjiiufioiis. ^31 whore /, and .are whole numbor.. and . i.slos. than .. Then, putting ( = c, the separate n.onibers of the oxpre.sMi.m j-^^, are • » B ' <^. "lo "nXnr;'"^:; T''"- ''^'--^--■'^;'="-"/^;; that is. the scoond te.mm(14),H,M,naltothe(.+ ir'tennin(ia). Again, if •>. = ,/. + . where '/ and . are whole nnn.bo,., ..„., „ i« ,«,, H.an ., 7^ = 7^ Also, because c.s = f,n-hc, w''=w'". Therefore »-«'yi;,;=,„'"/r -„."/'■. ff.nf; .i ,i • i tonn in (H) is e.ual to the (. ^ i).- tern. iJ^tS); al" l 'n.:!: ^ IT """^ Lot now s and a be two distinct value, of s, both less than u; and let ' »•«+ 1 = «•»+ 1 . "nd a-;+ , = a-^^ , . By what has been proved, the nun.bers . and . are determined by the equations e* = fm + r, la = iJn + z, h, ..,,,1 ^„ ,,ei„g mulliplc, „r „. Ji„t, ,i,„„ , „„d , „,, ,|i,|.„„, „„,, „ ■o«, ..■ and . mu,t l,„ din'oa.nt. Ilonco, a. ..,„ n„. ,hr„„sl, i„ „ vLi,,", to'.I;;o'"f ''■".""'"' ■■"" ""■°"*="' "' " ""'"'"'■ '^""■""y "1"»l. in .ome order! §13. From (5), i?,^ =ARf, at =B,Rf, where A,, B,, etc., are rational functions of tc'. Thesn valueq of 7?^ »^ .. substituted in (4), cause that expression to become ^" ^^" ''"' ^o" + w'R: + w"A,R: + w^'BX 4- etc. (15) Let the n values of the expression (15), obtained by putting t successively equal lo u, 1, J, . . . . I ,j — 1, be * ''li '2> • . . / I 'n Then, u being a whole number, 4 = ''e + U-b,R: + ?<,«c./ij; + 4- „y» -lV;r r, T.- • • • • . ^ e'-*'e } rl = «, + ^.-1^,72;+ M,-=c.,7?^+ . . . . + y:^^iif^ (16) n-l 232 Young : Forms, Necessary and Siifficknt, 0/ (he Roots 0/ where a,, h,, etc., are rational functions of w'. Therefore, if S„ be the sum of the V*'' powers of the terms in (10), S^ = nu^. liecause a, is a rational function of w% we may put na^ = J/ -f hw' + kiv^' +.... + /'m)<"~^'", where g, h, etc., are rational. But, by § 12, the n values of the expression (15), obtained by giving t successively the values 0, 1, 2, n — 1, are the same whatever value, making ic' a primitive h**" root of unity, be given to e. We may therefore substitute for iv', in the expression for na^ or S^, any one of the primitive Ji*" roots of unity w, to", v;'\ . . . . , w\ (17) Therefore S^=(j-{-hw + Jcid^ + etc. = g + hio" + hw^' + etc. =z (J -\- hw' -\- hii?' + aid. Therefore mS^ =m(j + ?i {w + 10' + etc.) + k{ic'^ + w'" + etc.) + etc., m being the number of the terms in the series (17). Consequently S„ is a rational and symmetrical function of the primitive n^^ roots of unity. Hence, by Kronecker's law, referred to in § 4, *S'„ is rational. This implies that the n terms in (16), which have been shown to be identical with the n values of x,+i in (2) obtained by giving s successively the values 0, 1, 2, . . . . , 71 — 1, are the roots of an ecjuation of the n^^ degree; that is, of an equation of the ?i*'' degree v/ith rational coefficients. Let this equation be /{x) = . § 14. In accordance with the proviso in 5 10, let the equation /(a-) = be irreducible. It is then a pure Abelian. For, taking rj, r^, etc., as in § 13, n = nf ■}■ R^ + ARt+ • ■ ■ ■ + <^e^/' A + Fji: + g,r: + . . . . + HX' (18) r'r' = Ke + LM + m,r: +.... + qm: where ^,, D^^ F^, etc., are rational functions of ic^. Multiply the first of equa- tions (18) by /*,, the second by Ji\, and so on, the last being multiplied by I^; then, by addition, ■ Kn + kA + .... + h>"r' = (kK + hD. + .... + hK) ^ + {K +hK-[-.... + hL,)R^ + : ; ^ Pure Uui-S'-nal Abcliun Equations. 233 Let the n - l quantities, h,, I;, etc., be determined by the n - 1 equations KA, + /.■,(?, + .... + IM^ = ,,,2«J^, Then 7,^,, + /,,. + ^to. = (/,,/^,f + Z- Z>. + . . . . + /,A; ) or, putting 7?'; for J^T?;-, and so on, ' ' ' Kn + /!v? + etc. = {h^El+ z- A + .... + Ik) + ^'^^i?:^+^^-4+.... + «;'»--4_... (19) «y § 12, a. +, -.-- .,^„ where r. = hn + c. When . = 0, c = 0. and when . = 1 c = e-, therefore " ^ , a-, = a|:^ -7+ Iit+ K+ etc. = r„ ''""^ ^'^ = «•«' + ! = Iit+ iifli^+ w-'-^Iit + etc. Therefore (19) may be written K^i + hA + etc. = {hM+ etc. ) - 7?; + a-,. But e may be any number that makes ..« a priuntive n^" root of unity, and (17) IS the series of the primitive ^^"■ ,-«ots of unity. Therefore a-2 = { RJ— (7,j/?.y 4- etc. ) } + h,x, + Z^rf +.... + .Vt? " ^ = li^o"— (Mo~+ etc.) }+Ji,x, + l'A + + ^,a-?-i = I i?o — (7<,i2; + etc. ) ! + 7*,xi + A-,:r? + .... + /^^n -i ^ where 7/^, ^„ etc., are what K becomes when to^ is changed into'«,, j.«, etc and ThP t'f v!'' n-'' ^' ^""""""' ^'''" ^''' '^ '^^""S^^ ^^*° "'- "'^ ^t^- -»d so on. Iheiefore, by addition, m being the number of the primitive n^h ^.^ts of unity mx^ = p + qxi + tx\-\- + vxl-\ where p, q, etc., are rational and symmetrical functions of the primitive «^" roots of , ty, and therefore are rational. Hence ., is a rational function of .,. Therefore the equation f(x) = is a pure Abelian. § 15. It is also uni-serial. For, by what has been proved, 234 YouxG •• Furm-i, Necefiftarj/ ami Suficienf, of ihc Booix of e.r, denoting u rational function of x,. But, from the form of x,+, in (2), since Bt= ^iHt, and Et= B,r}, and so on, we pass from x, to x, by simply changing /(•; into icRJ- The same change transforms a-j into x^. Therefore a-3 = 6xi = d'^xi . In like manner x,= 6h;, and so on, till ultimately 0"x,= x,. Thus all the roots of the equation f{x) = are comprised in the series x„ex„e'x,, e"-w Pure Abelian Equations ok Odd Prime Degrees. FmiUmmtal Ehmmt of the Root; the Root Gonstmcted from its Fundamental Element. § 16 We confine ourselves to pure Abelians of odd prime degrees, because the irreducible quadratic is always a pure Abelian. Let n be an odd ^prime number, and let the primitive n"' roots of unity be the terms w, io\ w\ etc., forming the series (6). Take ^, as in the first of equations (9); then, if R, be the fundamental element (see §6) of the root of a pure Abelian equation f{x) = of the n"* degree, it will be found that R,^Al^„ (20) yli being a rational function of xc. §17. From A, as expressed in (20), derive R,, R„ etc., by changing ic into ?i'», w\ etc. By § 5, the root of the equation f{x) = is R^-\-Rt+ R;+....+ Rl-i' (21) To construct the root, we have to determine the particular «*'' roots of Ro, Hi, etc., that are to be taken together in (21). When iv is changed into w% let A, become A,, as ^i becomes ^,. Then R, = A':^,. Therefore R: = w'A,^;, ^ (^2) .(/ ])eing an n»^ root of unity. In proceeding to make R: definite, we may first make <|); definite. By (9), ^ ^ Purf Uni-^erial AheUau Equntiom. to'' being an ;■ . oot of unity. Let A' p" p^ 2'65 (23) bo determinate; then, b, taking ^ ^ii,: Z'.^ nnir,. wo get ,f with ^ determinate value 0^"- {P\P\Pi ... p ) " Let us now consider d.,". Bv ^9^ >/)» hoino. ..,. ti> . .. TA ^y \^v), w Demg an u'" i-oot ot unity, ^di appeal in (23), they have ah'eudy been made definite We ff=-«^(p;p:,p\.....P„f, »; beh,, „„ „.» root Of unity Became . i. pH„e to „, the „-, ten,. ) tv * w , -J/i*" oro +V»/% «^ _ • . . to' , w'\ ?«", - - r -" », tuu n~i terms «> , are the same, m a certain order, with the terms , 10 . I herefore the term8 to, 10% to", . . Pj, Pfx, P'^ P^ may be taken to be the same, in a tertaln order,' with the terms in (23) They , Tf. ^^^ ^ "^ 5 = .^*^ (25) Re = A^; A. rega,* 4. we have B, = A;^,. But, by § s, ^. = P;.. Thet-efore ^f has a rational value. Consequently d has a rational value. In (21) ,„Wtute the rational value of H;. and the values of if, bJ. etc., given in (25). and the 236 Young : Forms, Necessary and Sufficient, of the Roots of root is constructed. In other words, the expression (21) is the root of a pure uni-serial Abelian equation of the «- degree, provided always that the equation of the n'^ degree, of which it can be shown to be the root, is irreducible. Necessity of the above Forms. & 18 The root a-, of the pure Abelian equation /(.r) = of the n'^ degree, n an odd prime, being assumed to be expressible as in (21), we have to show that its fundamental element 11, has the form (20). and that RJ, R:, etc., are to be taken as in (25), while /?t receives its rational value. §19. By (3), zbehi, any integer, FH being a rational fimction of lo. And equation (5) subsists along with (3) ; that is, e being any whole number prime to n, Ri=\F{w')\Rt. Give z here successively the values 1, /I, a, etc., these terms being the same as in the series (7). Then i?i= BJif, Ri= CM, Rl=zD,R:, £,, C,, etc, being rational functions of to*. Therefore {R'M.Rl..--Iie:r=(^M. where G, is a rational function of io\ and i=0 + e;i + ^a + + 0. From the nature of the series (7), = %-\ and . = X"-. Therefore .X = d. In like manner, each of the n - 1 separate members of t is equal to Q. Ihere- fore « = (n- 1)0. Because (6) is a cycle of primitive «»»> roots of unity, in other words, because ^ is a prime root of n, and = r-, i« P^'i"^/ ^« "; And n- 1 is necessarily prime to n. Therefore whole numbers h and k exist «uchthat /t< = &n+l. he same as Therefore Pure Uni-Serud AMian Equaliom. 237 Foy very integral value of . let (7..^)^ bo written pU then, putting i- for Hence, by puttmg c = 1 , and taking <^, as in (9), Thus^the form of the fundamental element in (20) is established. Also, when J^l = A{PlPiPl , , , , p^y ^ '^^-^^yi:^^,I^=A^. This is the first of equations (25). Since. r^Zri^''""^ to . let . = .. Then, from (.). becauL i = „ H Df=A{P:P:p',....p,)i. Therefore, giving . i„ (24) the value ,.. Itt= A J. This is the second of equa- tions (26). 1,1 hke manner we can show that all the terms E;, Jtf , . . ,{' are to be taken ae in (25). It has only to be added that sf mult bl token ivitl Us^ational value, because, by |5, uRJ is the sum of the root, of the equation Sufficiency of tJw Forms. §20. We here assume that ff, has the form (20), that RJ is rational, and *!' ta l2f obtataeTb'"'" " '° *'''■ ""' "° """'^ '° *"" "^^' ">= « ^-1"- of 3:,+ , m (2), obtamed by giving s successively the rvalues 12 „_, are the roots of a pure uni-serial Abelian equation of t'he'n- 'deVr^e pr'; vded always that the equation of the n- degree, of which they are the roots is irreducible. /„ ,ke first plac. 4 has been taken rational. /„ ,/„ njt Phee. an equal.on of the type (.,) subsists for every integral value of . n' lthV::ies^^'"^''"^°'"•, ';,""' r " ™"^^'-- '-^^ '» ^--"^- m me seues 1, 2, ,n~l. Then, by (25 j, {R,PT-f= {AAr%^^^r)\ (27) But 01 is the expression (8). Therefore, by § 9, 288 Young : Forms, Necessary and Sufficient, of the Roots of F{iii) being a rational function of w. This makes (27) an equation of the type (3). Next, let s be a multiple of n, in which case it may be taken to be zero. Then Therefore {l{.}{T')"=:Iif. • (28) Since /?„' is rational, (28) is an equation of the type (3), Therefore, whether a be a multiple of n or not, an equation of the type (3) subsists, hi the third place, the equation (5) subsists along with (3) for every value of e that makes ic' a primitive /t"' root of unity. For, let z be a multiple of n; it may be taken to be zero. Therefore 7?^ = BJ, and /(>; = l . Therefore /*! ,. ^- ,.- {liJK'Y^liS. (29) But, equation (28) being regarded as (3), (29) is (6). Next, let z not be a mul- tiple of n. It may be taken to be a number in the series 1, 2 ,n — 1. Then equation (27) is (3). But, in (27), z may be any number not a multiple of n, and ez is not a multiple of n. Therefore we may substitute for z either C2 or e. Thus we have {r,rt'Y ={AAT%<^,^rY. {R,,R7'Y = {A,,A-'){^,.^7'V . (30) But, equation (27) being regarded as (3), equation (30) is (6). Therefore, whether 2 be a multiple of n or not, equation (5) subsists along with (3). Hence, by the Criterion in §10, the n values of x,^^ in (3), obtained by giving s suc- cessively the values 0, 1, 2 n—\, are the roots of a pure uni-serial Abelian equation. Particular Valms of n ; tlie Pure Abelian Cubic. § 21. When the equation /(a;) = is of the third degree, taking ;i = 2, the series (7) is reduced to the terms 1,2, and the equations (25) become i?i = A, {P\P,)^ , Rl = A, (PIP,)^ . Also 7i'i = ^o, /•;, I and m are rational. Putting ^~ {P -\- ^rn' -\- lOlm)* _ 2 {P + 5m^ + lOlmf f^+5m=+2;m ' (35) (36) and (36) becomes or, putting 9 = p -|_ 5mi -I- 2(m P + 5m> + 2tm ' F{w) =p + q^z + V(//2 + hs/z). (37) Pure Uni- Serial Abtlimi Equations. 241 The value of . given above confon^s to the type (33). for it can be changed into Hence the general rational function of the primitive flfth root of uuitv falls under the expression for a, in (32). ^ § 26. The writer may perhaps be permitted to refer to a paper of hi. entitled jf: r^oT ll^t ^'T"'' ^T' !''''''-'' ^^"^'h'appearedtthi! mmu, vol. Ml, ^a. 2. ABsumnig llmt l|,e ,|i,hitic to I,e solved Im. l,v J,.,- he proved, in the ...tiole referred to. tlmt it admit, of algebraical solution o„l, i 16 -fl5^^~ P = and q = 16 + £> When the coefficients are thus related, take ?i a root of the equation X* ~ Bx^ — 6x* + 5x + 1 = 0. ^/(/~1) = — - . (lG + i?^X^. + l)(;2^1j» then the solution of the equation (38) is n = e^ + aQ^ + Xa'Q^— U'dK This form of the root may at first sight seem to have no affinity with the ^^^n 'T\' '"' " '^ — nication which was laid before' tr 1^. Soc ety of Canada at .ts meeting in May. 1886. and which is to appear nfhe orthcommg volume of the Transactions of the Society, the writer has 2wn the essential identity of the two forms. ""^^ The Pure Uni-Serial Abefjan Quartic. Necessary and Sufficient Forms of the RooUi. root AV;»i^'^'"^ ^ = ^ + 1 as in (33), the necessary and sufficient forms of the roots of the pure uni-serial Abelinn quartic are the expressions „,. „„ «,, „, t ^ \% 242 YoiNc; : Forwn, NarHsurii ornl SvffiaUnt, of the Hoots of (.T2); the rational expressions p,q,h,e being subject to the solo restriction that they must leave the equation of the fourth degree, which has a,, «,, 04 and «;, for its roots, irreducible. There is thus an intinuite relation between the pure uni-serial Abelian of the fourth degree and the solvable irreducible equa- tion of the fifth degree. This is only a case of u more general law. If 2n + 1 be any prime number, and if the forms of the roots of the pure uni-serial Abelian of degree 2/i have been found, the necessary and sufficient forms of the roots of the solvable irreducible equation of degree 2n + 1 can be found. ^Wffl>iitl/ of the. Forms (32). §28. Here an eciuation of the fourth degree /(x) = is assumed to be a pure uni-serial Abelian ; and we have to show that its roots are of the forms tt,, ua, (Xj, a,, in (32). The roots of the equation /(a:) = 0, in the familiar Abelian notation, are X,, ea-i, e^a-i, 0'xi. (39) Because xi is the root of an irreducible quartic, its form is where P is clear of the radical V Q. Another root of the quartic is P — V Q. This is obtained from x^ by changing the sign of V + Wv+V(/+'"V*) ^ ii Pun' Uni'Strial AMian lujuatiom. It (41) 213 th/; '■ ^"' /r'^ "u ^"''"^ '■''^'"""^- ''"J'^''^^"' ''■ P"««'ble, tlavt a-, is of the (ir«f, of the forma (40); then either •^-mei. oi or f'^i = /'-V. + V^.-.flV,=^, + V.+V^ = .-., or <'-^i = ^'-^^«--V/.-.0'x. = ;>H-v.+V/ = ;ri. But the equation /(.) = o, being a pure Abolian, is irre.lndble, and therefore cannot have equal roots. Therefore .. i. not of the Hr.t of the forms (40). IS therefore of the second. Consequently we nmy put a-, = y> + /.V.S + V(/ + wjV-v) I Oa-i =y, — /.V.v + V(/--»V«) I 6V, =p-}- /cx/h — V(/ 4- »j^«) [ 6\ =/» — /.V* — V(/ — »»-v/») J change tha causes a:, to become fl^., must transform 0V, into a-. We can now dete.-nnne the expression V(/ + »V.) n.ore definitely. To pass from .r, to Z we change the sign of V. and take the resulting radical ^{1- m^s) w U tt posmve sign. In order that these changes may cause Q., to become 6'.. the changes must admit of being made on .,. lu other words, the ^dic^ tt n7or?;^*^' ! , ' "'' °''"'' "^ *^^' '''^••'» '" •^•" •»"«* ^^« expressible in tei ms of the radicals m a-. . Therefore we must have s/{l - ms/a) = (c + (Ws) + (^ _ ,v*) V(/ + »V«) , c, d, gr and r being rational. Therefore ?-«,V.= (c + rfV.)«+(^_,V.)»(^ + ^^,) + 2(c. + rfV.)(^_,V6V(?+.«^.). Hence (o + dVs){ff - Ws) must be zero ; for, if it v.ere not, ^{1+ mV^) would be a rational function of ^s, which would make ., in (41) the rlt of a quad- ratic. And ff - rs/s cannot be zero, for this would make and therefore, by (41), 0^, would be the root of a quadratic. IS zero, and therefore V(? - »« V*) = (ff- WsW {f + mVs). ^^.^. By comparing the first three of equations (41) with one another, it appears that he change which transforms Vil+^Vs) into V{l-m^e) cau.es V('-^V.) tobecome -V(/H-mV«). Consequently, from (42), — V(/ + nl^/s) = {g + yV*') V(/ - ?« Vs) . (43) Hence c -f- d\/« (42) PI 244 Young : /iwtw*, y, ts,^, ami Sufficient, (rf the Rootn of From (42) aud (43), ^- A = - 1 .-. V" = ~'^~ ' (44) By 8(nmriii;; both siden ^^^ (43) and equating the j/arts involving the radical V*. li(// ' — ni(] + j/' + r"*). Therefore, by (44), 2j/f/=^ 2>» (1 + */»). ... Z = ^ (! + ^ii its value in (44). Then, writing 2 for 1 + {jj and '* f"r -' , and for ^9 Thus the necessity of the forms in (32) is established. Suffickncy of the Formn. § 'P"" We now take a,, Uj, a^, ug, as in (32), subject to the restriction that ihe quu .lO ec^i tion of which they are the roots must be irreducible, and we hav*^ to show that this equation is a pure uni-serial Abelian. The radical i^{hz—hy/z), which occurs in a,, is not found in that form in a,. But, keeping in view that 2 = e* + 1 , V(/»2-/lV2) = ^^^^^V(/*2+W2). (45) It is obvious that the expression ^ _ 5V2 + ^^^^ V(^2 + hs^z) C is a rational function of the expression p + q^/z + v/(7<2 + hy/z). Therefore Oj is a rational 1 unction of oj; and the equation /(a:)— is a pure Abelian. That it is uni-serial may be thus shown. To pass from oj to Cj, we change the sign ofv'z, and take the resulting radical \/(//^ — hy/z) with the positive sign. Let these same changes be made on a,. The result, by (45), is V2 + I p + qVz — 's/(/i2 — h»fz). And this again, by (45), is equivalent to p -)- qy/z s/{hz + hy/z), /*»*/e Uni-Serial Aheli l^intions. 246 which, becouse 2 = » • -f i , ia /■ + Qs/z- s^(hz + h^z),orai. nence. in pn«.sing from a, to «,. we pass fro/n «, to «,; ami in hko manner it may l.o shown that the same chnnges of the radicals carry us from a, to «, and from«3 back to a, ; consequently the pure Abelian equation /(x) = is xxni serial. ^ Tlie Funx4 + ic'xj = (x, - X,) -I- w (a-, - X3) . But, by what was proved above, a-i = i? + qyfz + 'sf^hz + /j^z), a:< = P + ^-v/z — V(/<2 + h>^z) , »*3 = i> — g-^/z — V{fiZ — /tVz). (46) Therefore, by (46), R\ = q»s/Zy 2Rl = v'(/»2 + ^Vz) + ws/{hz - h^/z). 246 Young : Forms, Necessary and Sufficient, of the Hoots of (47) Therefore, keeping in view that 2 = e* -}- 1 , and making use of the relation \/{hz + h>^z)\f{hz — /iVz) = he/>/z, 4Ri = h\^+l){we — iy B, = qV 4i?3 = /t»(e»+l)(iw + 1)» § 32. It may not be out of place to observe that, in (47), ^1 is not presented in the form in which it is a fundamental element of the root of the pure uni- serial Abelian quartic equation f(x) = 0; that is to say, it is not in the form in which 7?o. ff, and R^ can be derived from it by changing w into xc^ , 11? and xo^ respectively. Tn fact, by changing xo in U^, as given in (47), into v?, we should obtain -^ W((? -f l)(e + 1)»; whereas, by (47), R^ is 5V or q\&-\- 1)^ The form of i?j, in which it is the fundamental element of a^root of a pure uni-serial Abelian quartic, will be determined afterwards. The Problem of the Necessary and Sufficient Forms op the Roots of the Pure Uni-Serial Abelian Quartic Solved from Another Point of View. The Fundamental Element of the Root. % 33. The necessai'y and sufficient forms of the roots of the pure uni-serial Abelian equation of the fourth degree may be found in another manner ; namely, by making use of the principles laid down in § 5, so as to determine the funda- mental element R-^ of the root. Let to be a primitive fourth root of unity. Take any rational quantities, h, c, d, m. Find the rational quantities, p, q, r, s, by means of the three equations, equivalent to four linear equations, . p + q + r -i- s =d^ , 1 p — q -\- r — 8 nr {p~ Then it will be found that (t»+ c»)' . r) + w{q- s)=z -^^-jL Ri = p -{■ qiv -{• ru? + f>^i-^- (48) (49) w Pure Uni-Serial Ahelian Eqmtiom. The Boot Comtructed from its Fundamental Element. 247 into !o .1' ""' f ' " " ^''^' '''''' ^-- it ^0. i?. /?3 by changing -. nto n,o ^«^ «,3 ,espectively. But, since each of the expressions EIrI etc has our values for g.ven values of B,, R,, etc.. we must settle what lu of "^ expressions are to be taken together in order that Iil+R\+R\+Rl (50) may be the root of a pure uni-serial Abelian quartic. From the two equations R, = {p __ r) + w(q~s)=. '"'.(*_+^)' Bs={p~r)~w{q~s) = ""l^bz^l , i?ii?i = m. (51) Then, because R, = p + q + r + s = d\ take i?| such that ^0* = ^. (52) Finally, because R. = p-q + r-s = -^,^^ , let Rl be such that Rl is posi- 111 b'fthe'roTl" """' ^i'. etc. being thus determined, the expression (50) ntv^. , f ' P"'' "'''•''"'^ ^^^^'^" e^^'-^tion of the fourth degree provided always that the equation of the fourth degree, of which i cin be shown to be a root, is irreducible. ^" ""^ Necessity of the Above Forms. equali" /^:;r:;Tyi5" '^ '' '"' ^'^^^ °' ^ ^'--^ "-^-^^'-^^^ ^^^^^- ^-^ic ^^ = Bli-RJ+RURl, (53J and what we have to make out is that R, has the form given in (49), and that R. and i?j are related m such a manner that the equation (51) subsists while Ri IS essentia ly positive. When we say that R, has the form given inVL itl^ understood that,., ,, . and . are determined by the equations (48) ^ ^' §36. Because Fiw) in (3) is a rational function of w, we may put F(w) = (6 + cw)~\ M 24« Young : Forms, Necessary and Sufficient, of the Roots of (54) (55) h and c being rational. Therefore, from (3), taking 2=2, Ii\-{h^cw)-^Iil. Thei-efore, by (5), taking e = 3, R] ={b — civ)-^Rl Therefore Ri = {b' + c')-\R,Rs)h .-. i?, = (6^4-c^)-'(i?x/?3)i But ^1 is a rational function of iv. We may put i?, = < + tm; and i?3 = < — t^o, t and T being rational. Therefore R1R3 is equal to the positive quantity f + t*. Therefore, from the second of equations (55), R^ is positive. §37. Because b -f cw and i?, are rational functions of w, we may put {b + cw)-Uii=d + ko, d and S being rational. Therefore, from (54), R,= {{b + cw)-'R,\' = d^—h'+2dSw. Since R^ is rational, d6= 0. And 8 must be zero; for, if it were not, d would be zero, and we should have R^ = — h^, which, because R^ has been shown to be positive, is impossible. Therefore (6 + cMj)-»i?i = d) {b — cw)-'Rs = d) (56) Therefore also Therefore i?3^r'= {d{b + cwy\-^d{b' + c')}\ From (3), RgRr^ is the fourth power of a rational function of w. Therefore {d{b^ + c*)P is the fourth power of a rational function of w. Therefore ±d{b' + c') = {g + kwf = g'-L^+2g7cw, g and k being rational, the double sign on the extreme left of the equation indi- cating that it is not yet determined which of the two signs is to be taken. Hence gJc = 0. Therefore ±d{b^i- c^) is equal either to g^ or to — T^. That is, d{b'^ + c^) is the square of a rational quantity, with the positive or negative sign. Hence we may put + m* + sw\ which is the form of the fundamental element in (49). And, by §34, in con- structingthe root x^ from its fundamental element,^ having assigned a' definite character to 7?^ we then, knowing that RJi, is equal to m^ selected the value of 7?|so as to make R'iRl equal to m. Hence the necessity of the form of A in (49) and of the relation between the roots R\ and R\ indicated in (51) is made 250 Young : Forms, Necessary and Sufficient, of the Roots of good. At the same time, because Ii\R\ = m , R\B\ = m?; therefore, by the first of equations (55), Ii\ is positive. Sufficiency of the Forms. §.38. To prove that the above forms are sufficient, vsre have to show that the conditions specified in §10 are satisfied, it being assumed that the equation of the fourth degree, of which the root is given in (53), is irreducible. The first condition is that Rq must be rational. This is satisfied by the first of equations (48). Tlie next condition is that an equation of the type (3) subsists for every integral value of 3. It will be enough to consider two values of z, namely, 2 and 3. Because Ri=p + qto + riv^ + sio^ z=: {p — r) -\- w (q — s) and Ri=2^ + qid'+r + «?«« = (^j + r) — {q + s), we have, from the last two of equations (48), r,rt' = m' m" -j (i + civ)-* = (6 + cw)-*. (6» + c7 (6'-j-c')- Hence an equation of the type (3) subsists when 3=2. Again, -^8 =i' + qi(^ + rw^ + siv = {p — r) — w{q — s). m? (b -\- cwf But therefore Therefore {p — r) + w (q — s) — {p — r) — w {q ~ s) b' + c^ »l' (6 — CM))" 6' + o» ■ R,Rr'=m-*{h-\-cio)\h — cw)-*. Hence an equation of the type (3) subsists when 2=3. Consequently an equation of the type (3) subsists for every integral value of 2. The third co7idi- Hon is that equation (5) must subsist along with (3) for every value of e prime to 4. As we may leave out of view values of e greater than 4, we have only to consider the case in which e = 3. Also it will be enough to consider the cases in which z is equal to one of the numbers 0, 2, 3. Let s = 0. Then equation (3) is ^1 = \ F{io)\Rl = F{w). But R^ is rational. Hence, changing to into w", nl=F{w'). Also Rl=Rl Therefore Ri = Firv')=i{F{iv'^)]Rl. Pure Uni- Serial Abelian Equatiom. 251 NelV taT-? 't " """■ "''°" '="• '^'l''""™ (^' -I'™" »'»»« will, (3). SSQXX,, let 2 — 2. Then equation (3) is e v a ThP..f . • ''■ Ii^^-\FHVlil Iherefore, changing lo into id\ Therefore /^] = ..' { /'(-^) i /?|, t«' being an n*" root of unity. From (61) and (62), foremi^f^t!3T;'"'? Therefore ^(.3) = ,_,,. There- tore {F{w)}{F{w^)] is equal to the positive quantity g^ + h\ Also from the manner in which the root .. was constructed in ^ from its ^^^Zl^::, element. 7?,Z?|= „. Therefore (^7.3)^ is positive. Also, in constructing the root, m was taken positive. Therefore w' is positive ; that is, ^o' = U There- fore, from (62). 11]=. [F{r^)\Iil (63) But, equation (61) being (3), (63) is (5) ; so that, when . = 2, equation (5) sub- sists along with (3) Finally, let. = 3. Then equation (3) is ^ . . ^""^ = '?'^^^' (64) gi being a rational function of w. Therefore Therefore j3\ _ , . r,?. .« being one of the fourth roots of unity. From (64) and (65), (Ai?3)W)t«'=l. But in the same way in which the product of F(:^o) and F{v^) was shown to be pos.tive^,,y can be shown to be positive. Also {RJi,)^=m. Therefore (^ii?3) - m . Hence u/ must be positive. Therefore «/ = 1 , and (65) becomes . ^f = ?8^|. (66) Equation (64) being (3), equation (66) is (5). Hence, whether . be zero, or 2 or 3, equauon (5) subsists along with (3). Thus all the conditions specified in Li «rrA , ' ' ""'' ^^ '^" ^"^^™" ^" §1^' -^ *« the root of a pure uni-senal Abelian quartic. ^ 252 Young : Forms, Necessary and Sujfficient, of (he Roots of Identity of the Residts Obtained by the Two Methods. §39. It may be well to show that the results obtained by the two methods that have been employed for finding the necessary and sufficient forms of the roots of the pure uni-serial Abelian equation of the fourth degree are identical. In (47) wo have expressions for /?i, i?, and R^ as determined by the first method. What we need to make out is that these are substantially the san.e as the expressions for /?,, R^ and R^ obtained by the second method. By (48), Write -^ for and r> m' (6 + root of unity, w' a to", to'\ w"'' , w"'"- "I w\ W« , vo*" , 10° tO^*, t/;^*', , 10° (69) be cycles containing respectively all the primitive .»>' roots of unity, all the ZT^riT "[ ""'^' "^' '' °"- '^""^^ *h« --bers forming the series 67) be all odd each of the cycles (69) consists of more terms than one. Should the prime number 2 be a term in (67), say b, the last of the cycles (69) would be reduced to the single term < which it will be convenient to xegard as a cycle haTf is'l'tr'' 1 f' r '""• '" ^''^ ^^" '='' '' ^^y be assumed n «7 r. "': '^'" '' '"'^ '° "^ '' ''"-'''^' ^" the numbers ., t, etc., m (67) which are odd primes. The numbers ;i, h, etc., are prime roots o/ ., / e.c., respectively^ Take P, a rational function of ,, and, . being any integer Jet P, be what Pi becomes when 2 is changed into ^tf^ Put -^ & ' aK''' -* 3 ■* fl ■* fit jTfl Ph'-'- (70) In the case when one of the numbers in (67), say ft, is^2, the last of equations (70) is reduced to F = p . Then if E be the fundamental element of 'the root of a pure uni-serial Abelian equation /(x) = of the n»' degree, it will be found that , ^. . ^^ = ^i' (*'^r .... XiF^), (^2) ^1 being a rational function of w. ^ i iil 254 Young : Forms, Necessary and Sufficient, of the Rootn of (74) The Root Constructed from its Fundamental Elevient. §41. From 7?i, as expressed in (72), derive Hq, R^, etc., by changing w into w", w^, etc. By §5, the root of the equation /(x) = is R^ + /?; + /•'; + ....+ i4-i. (73) To construct the root, we have to determine the particular »t"' roots of R^, 7?i, etc., that are to be taken together in (73). When lo is changed into iif, lot -^1) 4'ii '^ii ®tc., become A,, ^,, i^*, etc., respectively. Then R.=A^,{^:A\ XUF!,)\ therefore E~: = w'AJ^^",,^:, .... Xt,F%f ] id being an ?t"' root of unity. Let the integers not greater than n that measure n, unity not included, be n,y, etc. (76) For instance, if n = 3 X 5 X 7 = 105, the series (75) is 105, 35, 21, 15, 7, 5, 3. The n"' roots of unity distinct from unity are the primitive n*** roots of unity, the primitive y* roots of unity, and so on. For instance, the series of the 105*" roots of unity distinct from unity, containing 104 terms, is made up of the 48 primitive 105*" roots of unity, the 24 primitive 35"' roots of unity, the 12 primi- tive 21** roots of unity, the 8 primitive 15*" roots of unity, the 6 primitive 7*" roots of unity, the 4 primitive 5*" roots of unity, and the 2 primitive 3'' roots of unity. The general primitive n*" root of unity being z«', give -d in the second of equations (74) the value unity for every value of z included under ^e. Then /^; = ^,(;, /?.;, etc., as in (76), (77). etc., we have determined all the terms , — J. ' Substitute, then, in (73) the rationaUalue which /?„^an be shown as in 5 8^'! possess, an the values of the terms in (78) as these "are determined i:(;:).\7'7) etc and he root is constructed; in other words, the expression (73 shall be he root ot a pure uni-serial Abelian equation of the «- degree, provided always that the equation of the .- degree, of which it is the root, is ineducible. Necessity of the Above Forma. /(x) of the n*" degree ,s expressible as in (73), and we have to prove that Its fundamental element B, has the form (72), and that the terms in ('T) Ll ^ be taken as in (76), (77), etc., while lij receives its rational value. § 43. By (3), z being any integer, Jfi = {F{w)\Ef, Fiw) being a rational function of w. And equation (5) subsists along with (3) : that IS, w' being the general primitive n'^ root of unity, ^ 4. = {F(w'),Iif. Taking 2=1, R^=B,Rp, B. being a rational function of w' . In like manner, taking z = X the'tt".' ' 'f '■'"""''"" '' "' ''^ '"^^ ''^y '' ^- ^« «howa that each of the terms in the series ^,.-, ^^'' ' ^""^ ' ^e,\ , i?,;,.-s is the product of i?. -- by a rational function of w\ Therefo «~XCXc;....i2„..-=)-=i.,z?, where F, is a rational function of w', and rf = Y =, GJi, {Rt'R', >£ 0i /?„^i»-0" = //«/?; (79) (80) (81) 256 YouNCf : Forms, Necessary and Sufficient, of the Roots of where (?,, H^, etc., are rational functions of itf, and (8 2) ■)^=qM (83) (84) From (79) and (81), where Q^ is a rational function of vf, and A is the sum of the terms d,h, ... that is, by (80) and (82), A = (J«(s-l);i'-» + T*(<-l)/i'-« + +/i»(i -!)/<;»-». Because i/3 = «=isa, and the prime numbers h and s are factors of n distinct from one another, h is a factor of a. Hence 6 is a factor of the first of the separate members of the expression for A in (84). In like manner & is a factor of all the separate members of the expression for A except the last. And it is not a factor of the last. For, assuming the prime factors of n in (67) to be all odd, since the last line in (69) is a cycle of primitive i"* roots of unity, h is prime to h. And h — 1 is necessarily prime to b. And /3 is prime to h, because /3 is the continued product of those prime factors of n which are distinct from ft. Hence /3*(ft — 1 )/«'-=' is prime to ft. The conclusion still holds if h is not odd, but equal to 2 . For, in that case, k—l and ft — 1 = 1 ; so that ^{h — !)¥-* = !5\ Now, /?' is odd, because /3 is the continued product of the odd factors of ??. Hence /3* is prime to ft or 2. Whether, therefore, the terms in (67) are all odd or not, every one of the separate members of the expression for A in (84) except the last is divisible by ft, but the last is not divisible by ft. Hence A is prime to ft. In like manner A is prime to each of the factors of n. Therefore it is prime to n. Therefore there are whole numbers m and r such that w? A =.rn-\- 1 . Therefore, from (83), (< . . . .)-(/??, . . . .) ;-. . . . (/2j;-. . ^.). =(g»./2,.)7i;;. For any integral value of z , let {B"^) " be written P; . Then, putting A Q^R't^ R'tAj^'xB the continued product of the expressions l-* er -r erft • • • • J^eth' ■) i 1-1 for I pi"-- pi"-' (pk'-''pK^-' Pure Uiii-Strial Abelinn Equafion'i. 267 (86) therefore, by (70), »r , , , i Thus the form of the fundamental element in (72) is established. Also, it was rrrtToTth'' ^"^^^;;'V^^--' -1-. ^---e. l>y §^- nf4 is the sum of which estabhshes the necessity of the forms assigned to all those expressions which are contained under lij. It remains t(, prove that the expressions contained under i?.;, |. or y being a term in the series (75) distinct from n, have the forms assigned to them in (77). §44. Since yv = n, and y is not equal to n, y is the continued product of some fthtt fof". 7' "':.\ '"' "'' '' *'^" '^"- ^^' ^' ^' «^-' b« the factors of n that are factors of ^, wh.le />, «f, etc., are not factors of ^. Because yv=.n = h3, andi,snotafactorof,;,^,isafactorof.. Let . = «i, then ./3=:i. The..! fore w"^ =z to'"" = lo". Therefore F„^ = /; ;. In like manner .r„, = A'o. And SO on as regards all those terms of the type F^, in which -f or b is not a measure of y. Hence, putting eo for . in the second of equations (74), and separating those factors of ^ that are of the type 1%, from those that are not, ^- = «^'^-(^W....)^(r..^I„....)^ (86) ^' being an n- root of unity. We understand that f/, Xf, etc., are here taken with the i-ational values which it has been proved that they admit. The con- tmued product of these expressions may be called Q, which gives us When e is taken with the particular value c, let «/' become ^«^ and when e has the value unity, let lo" become w". Then Because equations (3) and (5) subsist together, and «,« is included under < and (88) 268 Yoi'NO : Formal, Necenmry and SiipUnt, of th Hootn of (91) wliero /.•, is a rational fuiictioii of /r, and /•„ \h what k^ bocoinea by changing to into w\ l\y putting a = i in (a^), Ji^ - A,{tp'A\ . . . . XtFtf. Taking this in connection with the second of equations (87), In like manner, by putting c for r in (85), and taking the result in connection with the lirst of equations (87), iJUf7'f= »t"(^,„Jr")$(i^7'^ . . .) " \{rc..^7/'){^l.,^7r) . . . .}". (90) From (89) compared with the first of equations (88), and from (90) compared with the second of equations (88), l', = w"{A„Ar')Q{Fi"\. . .fm.r\ Therefore, because «t= «, W^<.^7'^f^/v, and y is not a multiple of b, v is a multiple of h. Therefore vjS is a multiple of i/3 or n. Therefore F^~ is a rational (92) 259 function of «■" [n \i\ra ~'' where i^, is a national function of ,k. in'Mko ^ being what, M, booonios in . his gives us ,. • , - "™"ios in passing from //•'• to //• lUr a , this last .Mjnution into „■'. This irivos ,.« * ^^ '''*' •"'" '"'""'go "" ''" Co Jquutions (8 7) becomes oforo which IS the form of 7?;, in (77). Sufficiency of the Forms. §«. Here „e «„„,„ ,h„^ j,^ „„, „„ ,„,.,^ ^^^^ ^^^^ ^ unkeri., Abelian equaHonVf the ^ del e Z"; r' '' "" '■°"' °'' " P"- which lias a rational value- and hv I, ,u- ,J- "'' wot of i?, rational value. /„ ,;,„ J^ j;:^ ™ ,1°^ "r .1" '" "'''' "" "'* *i, "'4';:r t T::::::^^r'':^-r. ,, . .. e„uati„r: -.' ^^ - --. P0«.. e? a ^a«ol;\:,arilr.I: J^^ ^ 260 Young : Forms, Necessary and Stiffident, of the Roots of rational quantity. In like manner A7« is the n"' power of a rational quant'^y, and so on. But Since each of the quantities i^/^, X^„ etc., is the n"' power of a rational quantity, let their continued product be Q'\ Q being rational. Then R.= {A.Qni>:.^l ). (94) Again, because zl^ is a multiple of n, Ff^ is the n*" power of a rational func- tion of w. In like manner Xf'^ is the n^^ power of a rational function of lo, and so on. Let {Ff'^Xf" ....)= iWf », Ml being a rational function of lo. Then R['=^ Ar\F^^'. . . .)(• power of a rational function of to, and so on. Therefore, from (96), B,Rr' is the n^^ power of a rational function of lo. This establishes equation (3) when a is the continued product of some of the prime fectors of n, but not of all. It virtually estab- lishes equation (3) also when z is prime to n, because this case may be regarded as included in the preceding by taking the view that the factors of n which measure z have disappeared. Thus, whether 2 be a multiple of w or be a mul- tiple of some factors of n, but not of others, or be prime to n, an equation of the type (3) subsists. In the third place, an equation of the type (5) subsists along with (3) for every value of e that makes 10' a primitive n*" root of unity. For, let z be prime to tj. It is then included in e. Also, since z and e are both prime to n, ze is included in e; and unity is included in e. But, from the manner in which the root was constructed from its fundamental element, i?; is determined as in (76). Therefore we have the four equations R;^= A,{^l,^l, ....Fl,f, li = A,,{^:,,^i ^/, Pure UniSerial Ahelian Equations. 261 (97) (98) Therefore '■>d (ff„ffr')-= (^„4-')(*„A--)- .... (F,„Fr,-)' . Because i^,,^--)'. „„d „,h„ eorresponding expressions have been shown to be a„„nal toeUons of the pri„,itive „•» root of unity „, the two eo.,ationr(97 correspond respectively to (3) and (5). If . be not prime to „, and yet not a multiple of ,., it may be taken to be ev, where v is equal to -"- , ,j being one "fvet'Ct I°f "'l"™!''^' .<"^"-' f™" ". «"1 «■ being the'general primi- means of (76), we can now, by means of (77), obtain the pair of equations (i?„7fr")^=(^,.^r")(^„,^-..) iB„,Br'r=z (4„^7")(*„„$r,") where ,o' represents any one of the primitive n'- roots of unity. Because such multiple of „, It may be taken to be zero. Then the equation corresponding to (^j IS, (^1 being a rational function of iv, i -^c' = qiRi ; or, since 2 = , i?; = ^1 . But BJ is rational. Therefore ,, is rational. Therefore ,, = ,,; in other words, q^ undergoes no change when tv becomes «^^ Also 4 = Iif= g^. Therefore since i?;=i, 4 = ?e/?J, Tattion f.^ T'l" ;°""PT'"^ '° ('^- ^^«-^-«- -batever . be, the equation (5) subsists along with (3). Hence, by the Criterion in §10 the xpression (73) is the root of a pure uni-serial Abelian equation of th; .- The Pure Uni-Serial Abelian of a Degree which is Four Times the Continued Product op a Number of Distinct Odd Primes. Fundamental Element of the Root. nrim^'^' ^^ ''"""' '"^"'"^ ""' '" '^' '""^^^"'^ P''^'^"*^^ of the distinct odd prime numbers, ,, * j j Take !' 'I' I- m 'i-mm^gmm.* ri 262 Young : Forms, Necessary and Sufficient, of the Roots of such that n = sa = tr = = i/3. Let w be a primitive n*" root of unity. Then w" is a primitive fourth root of unity, lo" a primitive «'" root of unity, and 80 on. Let ,,. IC M .»-3 J be cycles containing respectively all the primitive s*" roots of unity, all the primitive t'^ roots of unity, and so on. Let P, be a rational function of >r, and, for any integral value of z, let P, be what P, becomes by changing iv into of. We can always take P, such that P^ shall have the form of the fundamental element of the root of a pure uni-serial Abelian quartic ; that is, P„ may receive the form of B, in (49) as determined by the equations (48). For, because P, is a rational function of w, P^=a-\- aiw + a^n^ + + a„_iio"-\ the coefficients a, «,, etc., being rational. Therefore ■Pm = a + ajw" + a^w^'" + etc. = (« + «4 + etc.) + w^ (a, + fls + etc.) + ic"^ (a^ + etc.) + to^'- (a, + etc.) . This may be written Pm=/+fw^ +/"«'''" +/"<«'•". (102) All that is required in order that Pi may be a function of the kind described is that P^ in (102) be of the same character with E^ in (49). That is, we have to make /=p, f = g^f>^ ,., f" = s. By means of these four linear equations, the necessary relations between the quantities a, ai, a^, etc., can be constituted. Having thus taken P^ subject to the condition that P„ shall have the form of the fundamental element of the root of a pure uni-serial Abelian quartic, put «?. ■vL = P*' "p>i''~'p'>: •A' o-A' • l—t P..-^ 1 Tft' F, = Pi -pf-'pi II 'pk'-'pk'-' -» Si ■* fl 3*' (103) Pure Uni-Serial Abelian Equations. 263 Then if R, be the fundamental element of the root of „ n • • equation of the n- degree, it will be found that '"" """""'^ ^'^^^"^^ A being a rational function of 7,;. ^^^^^ The Moot Constructed from its Fundamental Element. . , §f • ■^''°"' ^''i' '^s expressed in (104) derive /? P . i. , into .00, .,3 etc. Then, assuming tha the roo nf fll' " " ^^ ''^""^^"^ "^ equation /(«.) = Q of the «- degrfe is ^"'' ""^■^'"''^^ ^^elian wh™.e to do in o^!:^^ [,, ^i;^;;; ^^^^^^ ^ By §8. *, is the «•'. power of a ™.il , <'»«) *: « the „.. power o^i™.!:," ::t'''t vi' ^''""'"■■^' ^=-- - = ". -ons ^;, J-S. etc., i. the „.^ powe'o Tationa "T' T' °' *^ ^"P-" or the »a.e ta, .Uh the fLa^enW "r: oTI" ^^^ „^ft' '"''"- ^' " AbeUan quartie, />. is ;ho fourth nowerof •, rn.lL l •""■" im.-serial « = 4« , /7 is the „- power of „ raZn^f ?•? ''T'"''- '''""•'"''>'■'=■ =»™ ^ the „. power of a rationa, tT^^^^^' ^''T "'"' ^''"^' ''' Sfio T at tur. , 4"«i"i.uy, and li„ has a rationa value ^ § ^0. Let the „„„hers .,ot o.ceedi^g n that „,eas„re „. unilr^o. ;„,„,,, n . '*' y- etc. For instance, if . = 4 x 3 x 5 = 60, the series (107) is ^'''^ ThP t. , . ''■^^'20.15,12,10,6,5,4,3,2. Ihe w*" roots of unitv distinof f.-^r^ •. of ™ity, the pritnitive ,- ots oZrir.T """' "i^ -' '"^ P™'*'™ "" -'» the fifty.„i„e »» roots o( u„ ty dilt „ fr^ T' ^" '"'"'"°^' "''™ » = ««■ roots of „„ity, and the eigh prt "ve 30 " ""' 7" ""° ""'"" P'™'"'« «»'^ tive 20". roots of unity, a'd th ^ p ■ „!»- 1" ""?' T" '"^ ^■«'" P™'" pnmitive 12- roots of uuity aud thTT ''"''^' '"■* *« four two primitive 6- roots o u„i y .1 "J f'""'™ ■»" ■■oote of unity, and the the two pri„,itiye 4" roots of tmit; a„d JT """"'"' °" """' »' "■'■'^. -•! unity, and the two pnm.tiye 3" roots of unity, and 264 Young : Forms, Necessary and Sufficient, of the Roots of the primitive 2'' root of unity. According to our usual notation, let Pz,^z, etc., be what Pi, «?)i, etc., become when %o is changed into «o*, a being any integer. Then, from (104), l^^ = A^.^P^n^^M Ft^) ^ ^^^^^ Therefore /?; = idAlPl,.^lMr .... Fl^ " 3 y,^ being an n"' root of unity. The general primitive n"' root of unity being 7t«, give v^ in the second of equations (108) the value unity for every value of z included under e. Then ^ i?; = A(Pr.^L^;,....n)". (109) Taking any number y distinct from n in the series (107), since v/ is a factor of n, jgt y^-ra. Then nf is a primitive y*"^ root of unity. Hence, since i^' is the general primitive n"' root of unity, all the primitive y'-'^ roots of unity are included in iif\ If ^d in the second of equations (108) be t^" when z^v, give •? roots of unity dib inct from unity is made up of the primitive n*'' roots of unity, and the primi- tive 2,*'' roots of unity, and so on, all the terms 1, 2, . . . . , n- 1 will be found in the groups of numbers represented by the subscripts c,cv, etc., when multiples of n are rejected. Consequently, in determining P; , P.T, etc., as in (109), (110), etc. we have determined all the terms Rl,R i I , Rn-l' (111) Substitute, then, in (105) the rational value of Po", and the terms in (111) as these are determined by the equations (109), (110), etc., and the root is con- structed; that is, the expression (105) is the root of a pure uni-serial Abelian equation of the n*^ degree, provided always that the equation of the n"' degree, of which it is the root, is irreducible. (111) P^n-e Uni-Serial Ahelian ^nations. Necessity of the above Forms. 265 -'o. exclusive of ,^. The terms ,r ,.■" «. ■ ''i' "" ""'" <=»™ponding roots all the primitive .-» roots of ^^, \ 7.:^ ZtT "' "' "'*' """' '°'"« 9 0^. llie general pr m tive n^^' mr.^ ^e -x , . leaving distinct residues^hen I ,1?/;: ' '""^^^" ' ^-^-^-sof ., form multiples of s are rejected, can be found of the .being a whole number. For, «i„ee »=,, the ,- l terms "+'•^ + 1 («-.) "• There- are primitive „•» roots of unity. This implies that ther, e are s—i values of ^ 266 YouNO : Forrrifi, Necessary and Sufficient, of the Roots of in (112), zero included, which make w'"'+^ a primitive »*" root of unity. Let two of these values of g be gi and g^. Put ■ become, lo"-- but bv Therefore, from (117) and (118), a,.6- + «.«:- + etc. = u„r" + ,,,.- + etc. (no) ^«" («i - «,) + «;-' {a, - «3) + etc. = . P^ieTtha'f '/'' "'"^^^"^^ ^'-^'^' ^'^-«^' etc., must all vanish. This im- plies that «„ «, „^_^ ^,3 ^^ij ^q^^^, ^^ ^^^ ^^^^^^^ ^^^^^^^^ ^'^" = «„ + «,(«,' + .„^^4. etc.) = .„_,,. ^j Thus 7, . clear of .. m like manner it can bo shown to be clear of all the i',.^^--..g what ^. become, when «, « changed into «-. These are the equations • §M. From what has been established, it follows that It has the form of he fundamental element of a pure uni-serial Abelian auartic." ZZ § al h s requtred ,n order that B may have such a form is that the equa In (114) should 8ub8,st, and that i?] should have a rational value. By 85 IccT .a the fundamental element of the root of a pure uni-serial Abelia' oquZ !? the »". degree, B; has a rational value. Therefore III has a rational va.ue. i 268 Young : Forms, Necessary and Sufficient, of the Roots of §56. In the very same way in which (83) was established, it can be proved that JfUJ^CK': .... /O =)" • • • • (^'V ..••)■"= QelfJ, (121) where Q^ is a rational function of w', and A = m»-|-a«(8—l);^'-»-fT'(<-l) /*'-'+ +^'{b— 1)/^;*-'. (122) Because ?« is the continued product of the odd factors of »i, ??»" is odd. But each of the expressions s — 1 , ^ — 1 , etc., is even. Therefore A is odd. There- fore A is prime to 4. Again, because m is the continued product of the odd factors of n, it is a multiple of h. And, because sa =-h^, <7 is a multiple of h. In like manner t is a multiple of h. In this way all the separate members of the expression for A in (122) except the last are multiples of h. And, by the same reasoning as was used in §44, /?^ (i — 1)//"'' is not a multiple of h. Therefore A is prime to h. In like manner it is prime to s, t, etc. Therefore it is prime to n. Therefore there are whole numbers v anu ;• sucii that v A = rn + 1 . Therefore, from (121), ^ 7C(iC". . . )''{iC'- . . .)^. . . . (i?j; . . . •Y={Qimii:- (123) For any integral value of z, let i?; be written P;. Then, by (103), putting A-^ for Qlir,, (123) becomes Therefore B^ = A^P^l^M ....Fi). (125) But P„ is the same as R'm. Therefore, by § 54, P„ is of the form of the funda- mental element of the root of a pure uui-serial Abeiian quartic. Therefore the expression for i?i in (125) is identical with that in (104), and thus the form of the fundamental element in (104) is established. Also, it was necessary to take i?o with its rational value, because, by § 5, nRg is the sura of the roots of the equation /(a-) = 0. And equation (124) is identical with (109), which estab- lishes the necessity of the forms assigned to all those expressions which are contained under R^ . It remains to prove that the expressions contained under pj;, _ or y being a term in the series (107) distinct from n, have the forms assigned to them in (110). The details to be given here are very much a repe- tition of what is found in §44; but, to prevent the confusion that might arise Pwe UnirSerial AUlian Equations. 269 of n of wl,ich „ i. a ™,;,,^ L ':J '■ ^°' •■ '■ "'"■■ "•" ""= '"'<' f-'»- i» a facto. Of „. Let . = „,, tle„ ^ = t " ,e eLl' V !:7 "/Z; '' »«".- ^,. = X., and «, on as .ga.d, a,, tho. te™/':f-thtt, ^ f' - ""..ch ^ or * i, an odd factor of „, but not a factor of ,. Hence, putting I for . ,n ,ho .econd of equations (108), and separating those factors of li^ that are of the typo F^, from (h«e that are not, ^:; = . ^'^^-> because where ^, ,s a rational function of w, and Z- is whnf 7- i into .^ By putting . equal to unit; "(10^ """ '^ ^'''^"^^"^ ^ ^''i" = A, (P„"|0j .... 7P|)i^ Taking this in connection with the second of equations (l 27) 270 Young : Forma, Neaissarij and Sufficient, of the Roots of (131) (132) In like manner, by putting c for e in (109), and taking the result in connection with the first of equations (127), {n,„RZ^Y=unA,A7'')Q{Fr^'- . . .Y\{P7..P7D{rn.^7,'') . . . .}"'. (130) From (129) compared with the first of equations (128), and from (130) com- pared with the second of equations (128), and l'„=icr{A,A7'')Q{I^V'' ■ • -f \{PT.n.P7r){¥o^,^7J'-) ■■••] Exactly as in § 44, it can be shown that {¥c.a^7,'"Y=qc, (/„ being a rational function of the primitive ?«*" root of unity w"^. Also, it has been proved that P^ is of the form of the fundamental element of the root of a pure uni-serial Abelian quartic. Therefore, by (3), {PcvmP^Y is a rational function of the primitive fourth root of unity w"*. Therefore, because n = 4»i, {PTvmPm'")' is a rational function of the primitive n"' root of unity w". Put {PT.mPT^'^f = q'c- (133) Again, exactly as in §44, F,„ ' ^q'J, (134) q'J being a rational function of ic°. By (132), (133), (134), and other correspond- ing equations, the second of equations (131) becomes h, = io^{A,„A7^)Q{q,qlq'J..,.). (135) In like manner, from the first of equations (131), /.i = i«»(A^r"')<3(Mi'?('----). <7i, q[, etc., being what q^, ql, etc., become in passing from w" to iv. It may be noted that this assumes that we are entitled to change equation (133) into /■ Dm TJ— t)m\» -^ „/ K-l^^vm^m ) — qi- vt_ in_ m^ '.11 The warrant for this lies in the fact that the roots Pl^ , Pj"^ , Pg^n , or P,5i . Pim . Psm i were taken with the values they have in the root Pl+Pi + PL+PL of a pure uni-serial Abelian quartic. This being so, the equation (Pr.P;;™)^= " _ aaA ^,t. ■ v^-'V -™e „.4u^o^ ;;ici:,r ;,;r;i:r'r '° "" which IS the form ofiC in (110). ions Svffioicnnj o/thr. Forms. pure „„«erial Abelian equaio„'„f ,,'",' ""P""'"" (•»•'' '« '^^ '«»' of a For y, „-._,, !- ° r ' * """"" *"• """'-^ """S-l value of .. ^oii-r;:;: Vetjr:;;':' rr- f t r r - -^ -■ rational function of the primitive fouM, J.S'f" ; °'""' P"™"- »f " proved, exactly a, in §44 that w' the T """"°" "' "' ^'"O' " «'■' be »•» power of : ratioL.'f:f„:i„?:;': ; 'i:rf:tT v' "■■''" '- ^"^ expressions. Therefore ii! fi-- i. n,„ ii, "'""' otter corresponding »e aMj,bce. „e have tofh'o ' h r '""" "' " ""''"""'' '""""o- ■>'"■ I correspon'ding equat 3, Ir fet "Ar^"'"""-* - W -^-'ts for every Also, since . and . are bo h pll I „ """". \ "l " '» ""'- ^d'-^ed in . eluded in .. But, from th 7 Zr „ whil';: ^ " ^^ ""' """^ " ■■"■ f„,„l . , , ■ ^° ■'»°' ""^ constructed from its fundamental element, />•; is determined as in (109). Therefore 272 Young : For)»'<, Necesaani and Snffinertt, of the Roots of (130) /?;= A (/':«?>: ....^1)",. Therefore (/^/?r') "' = (^.^rO (/'m/V/f («4'"' and {nj^7'f = (^..^r')(/'«-n^''f (^'-'/'^ Because (P, P^')^and other such expressions have been shown to be rational fun.'tions oMhe prinutive n'»> root of unity, the two equations (106) correspond respectively to (3) and (5). If z be not prime to n, and yet not a multiple of n, it may be taken to be .v. where « is equal to ^- , y being one of the terms in the series (107) distinct from n, and u- being the general primitive »'•' root of unity. Then, just as we obtained the pair of equations (136) by means of (109), we can now, by means of (110), obtain {l^Ji^^^f = (^.„^r"') (P.m^rf . • . • I (137) {lUJ^*'f = (yl„„i7"')(/^c.mPr."')^ . ■ • . ) where .•« represents any one of the prinutive «'" roots of unity. Because (P p-'"f and other such expressions have been shown to be rational functions of The primitive n'" root of unity, the two equations (137) correspond respectively to (3) and (5). Finally, should z be a multiple of n, it may be taken to be zero. Then the equation corresponding to (3) is 5i being a rational function of w . Or, since z = , But lit is rational. Therefore q, is rational. Hence, if q, be what q, becomes in passing from ^v to ^.^ ?. = q. Also 4.= Bj = ./.. Therefore, since Pj = 1 . Iii=qjii, which is the equation corresponding to (5). Therefore, whatever .be, the equa- tion (6) subsists along with (3). Hence, by the Criterion in § 10 the expression (105) is the root of a pure uni-serial Abelian equation of the n degree. I:« Pure (/ni-Seriat Abtlinn lu/itutiwi^. Ror-VAnM.: Ihukwcwuk KytfATioxs ok Puime DEanm-.s. 273 l^vo.uf ,t bo not a pure Aholian. tho uocoHHury and siiffi.-iout Co rms of itt ! " can. by .noann of the ,.robIe.ns solved above, be doter.nine /„ / T-l u^eif/u the contmmd product of a numUr of distinct prime, or four fL i ront.nu:d product of a number ,^ distU^t odd prim,.. ' ''""' ''""' '^" §5y. It ,s kriowu that tho root of tho equation is of the form where k is rational ; and t" ^^n-i, (j.^g^ ^^>'^''»' ^»-«. (139) are the roots of an equation of the n'" decree that is of „n n r rational coefficients. Let thi« equation be ^ (. = T e plrorti. T /(-) = may also be expressed in the form ' " "'""''"" Where «!, i,, etc., are -ational functions of // ^ ^ exp.,.„„ (HO, „..o ..can. „,,„, .r:!r\.!t;r: (u™xr° Thereforo li=a,n'. l,.,.„ce, since ». i, „ rational function of ft /, i, ' mional funct.on ot S,. The expression ft is tl„.s the root of l^e It eates the degree of the equation, and is therefore our " wh le" iS O T - quantities involved rational,, in the coefficients of he euattn A.) -t' Hav.ng g,ve„, after Abel, what are subs.antia.ly the two hJ^nZi.7,7l p':;:::;;^;. "iz '"- -"' "-^ "-"^ '--"o- -^^^^ pioDierae propose, doit pouvoir se mettre sous ces dpnv fnrmoa ^nt encore trop g,„,r.|es, e'est-Vdire qu-eLTrenJerlT sZZtr etj a, t,ouve dabord que parm, Icsfonctions re„fermces dans lu forme ^2) » [the 274 Young ; Forms, Necessary and Sufficient, of (lie Roots of same as (138)] "celles qui satisfont au problfeme propose doivent avoir la pro- priete nonseulinent que les fonctions symetriques de R^, R^, etc., soient ration- nelles en ^, B, C, etc. (ce qu'Abel a remarque), mais aussi que les fonctions cycliques des quantites Ri, R^, etc., prises dans un certain ordre, soient egale- ment rationnelles en ^, B, C, etc. ; en d'autres termes, I'equation de degre ft — 1 , dont Ri, R2, etc., sont les racines, doit etre une equation abelienne. J'entendrai toujours ici par equations abeliennes cette classe particulifere d'equations reso- luble qu'Abel a considerees dans le Menioire XI du premier volume des CEuvres complUes, et dont je supposerai les coefficients fonctions rationnelles de A, B, C, etc. En designant par Xi, x^, x„, des racines prises dans un ordre deter- mine, ces equations peuvent etre dofinies soit en disant que les fonctions cycliques des racines sont rationnelles en A, B, C, etc., soit en disant qu'on a les rela- tions, Xa = 0(xi), a-3 = 6(a-j) , -Tn = ^ K-i), a-j = Qx^, oil (a-) est une fonction entiere de x dont les coefficients sont rationnels en A, B, C, etc." In saying that the y. — 1 (or, in our notation, the n — 1) terms, i?i, i?2, etc., are the roots of an Abelian equation, Kronecker must be understood to assume that the equation 4) (x) = , which has the terms in (139) for its roots, is irreducible. As a matter of fact, in the most general case, which includes all the others, the equation «^ (a;) = is irreducible. But in particular cases it may be reducible, and then it is not an Abelian. In a paper by the present writer, entitled "Principles of the Solution of Equations of the Higher Degrees," which appeared in this Journal (Vol. VI, No. 1), it was proved that when the equation ^ (a:) = is reducible, it can be broken into a number of irreducible equations, ^i(a:) = 0,'4'2(a;) = 0, ^,{x) = 0, each a pure uni-serial Abelian. Hence, for a detailed discussion of the problem we have now before us, we should require to deal not only with the general case in which the equation 4) (x) = is irreducible, but also with the several, cases in which equations such as •4'i(a;) = 0, •^^{x)=.0, etc., can be formed. But since, as has been stated above, the particular cases are included in the general, we shall confine ourselves to the problem of the neces- sary and sufficient forms of the roots of the solvable irreducible equation f{x) = of degree n, when the subordinate equation ^(x) = of degree n — 1 is irreducible, and is therefore a pure uni-serial Abelian ; it being understood that n — 1 is either the continued product of a number of distinct primes, or four times the continued product of a number of distinct odd primes. Pure Uni-Seriul AheUan Equations. 275 Form of the Hoot. §60. The solutions of the problems investigated in the precedinir mrt of the paper have furnished us with the necessar/and sufficient'for n I oo o he pure un.senal Abelian equation ,(.) = o of degree .- i. Let this »'l. n, 'a. .... , /•,, Vg. ^l^^\ It Will be found that the ter.s 4, 4, etc., in (138), which are the same, in a cei^a,n order as 1^[, /.- , /^ . etc., with multiples of n rejected from the sub scripts, are given by the equations 4=A4ry,r:.. 4 = A^{rtry„.. (145) H; = A, {.[ri .... rlf ^nnl^^ t\^«"bscripts of the factors of the expression for JRjAr^ are the terms er es 'I ' '';^"'"" ^" *'^ ^"'"^^ ^" (^^^) "^ --rse order. Because t" Necebsitt/ of the above Forms. §61. Here, »ssumi„g that the root of a solvable irreducible equatio,, of tZTiuir'^'""" " '" <''"■ "" ""^^ "> *- ">»' ^.'. a'. ■='-. have the §62. In (138) Itf is an «■> root of J}„ one of the roots of a pure uni^eri-,1 Abehan equation * (.) = o, the series of whose roots is contained in (13;). Zt !■! 276 Young : Forms, Necessary and Sufficient, of the Roots of El may be any one of the roots. This implies that if the roots, in the order in which they circulate, are El, B^, Ba, . . . • . Rti ^,y ^*'». the change of B^ in the system of equations (141) into BH will cause J?; to become B^, and B^ to become B; , and so on. In fact, by exactly the same reasoning as that used in establishing the Criterion of pure uni-serial Abeli- anism, it can be made to appear that the n values of the expression (138) or of (140) obtained by taking the n values of B^ for a given value of Bi, and X i, taking at the same time the appropriate values of i?; , B^ , etc., as determined by the equations (141), would not be the roots of an equation of the n^^ degree 1 J. with rational coefficients unless Bj couid replace i?; in the manner above indi- i_ cated. In like manner, by changing 7?i" in the system of equations (141) into B^, jR;^ becomes Bj, and so on. The principle can be extended to all the terms in the series b;,b:,b: b:,b;. §63. Let, then, the system of equations (141) be written Bi = a'Jif, bI = b'Jif , etc. , (146) (147) e being a general symbol under which all the terms in the series (143) are con- tained, while ai, hi, etc., are rational functions of B,. These equations give us (BtBl^Bl .... B%B'„B./' = gM, where G, is a rational function of B^ , and t = e-\-eX + ba+ ... . + e = (n-^l)0s=(n-^l)r-*. Because /I is a prime root of w, (n— 1) X""* is prime to n. Therefore t is prime to n. Therefore whole numbers h and 7c exist such that Therefore ht = 7m + 1 . {BiB[,....B,,y={G':B.'^)B:. For every integral value of z, let {B",,)" be written r,;. for G'^M, j^i ^ j^ (^j^.^,«_^ __ ^,,..^^)i. Then, putting Aj^ (148) Pure Uni-Serial Abelian Equatmi^. 277 Because /•„ is simply another way of writing R%, and the terms B„ li, etc are the roots of a pure uni-serial Abelian, it follows that /•„ /•„ etc, have the forms of the roots of a pure uni-serial Abelian. By putting e, then, in (148) succes- sively equal to 1, X, a, ..... 0, the ti- 1 terms in (146) are obtained with the forms assigned to them in (145). Sufficiency of the Formal. §64. We here assume that the terms forming the series (146) are taken as m (145), and we have to show that the expression (140) is the root of a solvable irreducible ecjuation of the «»•> degree ; provided always that the equation of the n"' degree, of which it is a root, is irreducible. Because the terms forming the series (146) arc aken as in (145), the system of equations (147) subsists. Therefore, by a course of reasoning precisely similar to that used in an earlier part of the paper to show that the r> values of the expression (2), obtained by giving .s- successively the V!».'!>i,i 0, 1 , 2 , , « — 1 , are the roots of an equation of the «*» degree, it can ; ov.' :.,e shown that the n values of the expression (140), obtained by taking the n values of i2,~for a given value of R^, are the roots of an equation of the «'»' degree, that is, of an equation of the «»•> degree with rational coefficients.