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Biquadratic equations with prescribed groups Moyls, Benjamin Nelson 1941

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BIQU-ADRiiTIC EQUATIONS WITH PRESCRIBED GROUPS by Benjamin Nelson Moyls A Thesis submitted in Partial Fulfilment of The Requirements for the Degree of MASTER OF ARTS in the Department of The University of British Columbia September, 1941, • i -T A B L E OF CONTENTS Page Acknowledgment......................*................ i i X * X^itr0ClVl0t*i.O21» « « « » » * e * « * 9 * » « * « • * * « • • • « • • * « • • • » « • • ? • • * • X XX » Ttl© Gr3T OVtpS • • » • • * • • » • • • « • • « ..» • « « « « * * e * e ft • • » « e « » « » » « « « « 0 ^ EquatIons with the Quartio Group......... 5 Equations -with an Ootio Group...,,..,...., 9 Equations with the Alternating Group..... 12 Equations with a Cyolio Group...........* 15 Bibliography . • 20 III. Biquadratic IV. Biquadratic V. Biquadratic VI• Biquadratic - i i -fh©; w r l i i r wishes to express his thanks to Dr. Ralph Hull of the Department of Mathematics at the University of British Columbia for his advioe and guidance* His numerous oritioisms and suggestions proved invaluable in the pre" paration of this thesis. Ben Moyls Sept. 1941. BIQUADRATIC EQUATIONS WITH PRESCRIBED GROUPS I. INTRODUCTION. One of the problems of the Galois theory of equations is that of determining values in a f i e l d F, of the coefficients of an equation x-t-a.x +—-» +aK=o, (1) such that the equation w i l l have a specified Galois group with respect to F. In seme cases, at least, i t is possible to find parametric repre-sentations of the equations having a specified Galois group. This means that the ebeff icients a t , a ^ , oan Me expressed i n terms of para-meters such that, when the parameters are assigned values i n F, the resulting values of a,,---, a^. are the coefficients ©f an equation whioh has the speoified Galois group with respeot to F, and with the further property that a l l such equations can be so "obtained. The values whioh may be assigned to the parameters are usually not completely arbitrary, but are subjeot to restrictions which ensure that the equation shall actually have the specified Galois group and not a sub-group of i t . In other cases i t i s convenient to employ two or more different sets of parameters, corresponding to different forms for the coefficients, which together give the totality of equations having the speoified group. Seidelmaim ' has found paramefcrio representations for cubic equations with the oyolic group, and for biquadratic equations with various groups. He introduced appropriate parameters into the forms taken by the roots of the equations, according to the Galois group of the equation, and calculated the forms of the coefficients from these. Another general method of attacking the problem, called the "rational" method in contrast with Seidelmanns "irrational" method, has been discussed by F r l . E.Noether )^ . She showed that the existence of a parametric form for the equations with a specified Galois group depends upon the existence of a minimal basis for the rational functions of n variables which are unaltered by the specified group of permutations of the variables. A minimal basis for suoh functions belonging to a group, or, briefly, a minimal basis for the group, is a set of rational functions of the variables which are: (a) unaltered by the permutations of the group, (b) algebraically independent, (c) such that every rational function of the variables which is unaltered by the permutations of the group is expressible as a rational function of the basis functions. For example, the elementary symmetric functions of n variables form a minimal basis for the symmetric group of permutations of the variables. Seidelmann's results indirectly 1) Seidelmann, Fj "Die Kub. und Biquad. Gleiohungen mit Affekt" Math. Anna1en, 1918, pp. 230-233. 2) Noether, E: "Gleichungen mit Vorgeschriebener Gruppen" Math. Annalen, 1918, pp. 221-229. - 3 -yield minimal bases for the groups he considers, and Breuer > has determined minimal bases for certain metacyclio groups. It is the purpose of this thesis to apply the rational methods suggested by Noether to the determination of the reduced biquadratic equations x^rf a^x i+ a 3 x-t-a.tf.-Q, (2) having speoified Galois groups with respect to the f i e l d E of rational numbers. The groups to be studied are described in the next section. For each group in turn, a minimal basis is then exhibited, the coefficients of the equation are expressed in terms of the basis functions whioh serve as the required parameters, and then restrictions to be placed upon the values in R, which may be assigned to the para-meters, are studied. Since the existence of one minimal basis for a group implies the existence of infinitely many, a variety of para-metric forms for the equations can be obtained. The general plan has been followed of using a minimal basis which seems likely to yield especially simple parametric forms for the coefficients of the equations. In some cases alternative minimal bases must be employed, in order to obtain those equations having the specified groups which correspond to exceptional values for the parameters" f i r s t employed. Certain of the minimal bases used were suggested by Noether, that for the cyclic group by Dr. Hull. The results obtained are equivalent to those of Seidelmann, although in a different form whioh is simpler than his in some oases. 3) Breuer, S: "Matazyklisohe Minimalbasis und Komplexe Primzahlen" Journal fur Mathematik, bd. 156, 1927, pp. 13-42. Certain modifications of the forms would be required i f a coefficient f i e l d other than R were used. II. The Groups. The Galois group of an irreducible biquadratic equation, with respect to the f i e l d R, is a transitive group of permutations on four variables. We confine attention to the equations whioh are said to have an "affect", that i s , have a Galois group other than the symmetric group i t s e l f . We denote the variables, and the roots of a biquadratic equation by x 0, x, , x t and x 3, and a permutation such as (x„x (x x) by (012), as usual. The groups to be oonsidered are as follows: (a) The Quartio Group (Die Vierergruppe): Q : [ i , (0l)(23), (02)(13), (03)(12)] (b) The OotiOiGrdups: 0 : [Q, Q(13)] 0': [Q, Q(12)} 0": [Q, Q(23)] (c) The Alternating Group: A s [Q, Q(123), Q(132)] (d) The Cyclic Groups: 0 : [ i , (0123), (02)(13), (0321)] C': [ i , (0231), (03)(12), (0132)] C*: [ i , (0312), (01)(23), (0213)] There is no loss of generality or completeness i n dealing only with the groups Q, 0, A and 0, since the groups o' and 0" are equivalent to 0, and correspond merely to a re-numbering of the variables, and, similarly for C , c' and c". It i t to be noted that C is the oyolio subgroup of order 4 of 0. III. Biquadratic Equations with the Quartic Group. A basis for the Quartio group can be constructed as follows Let 4u„ =s x0+ x,+ x^+ x 3 4u, = x^x^-x^-x3 4u z = x,~ x,f x^~ X3 4u3 (3) Then x 0 = u +u,+ u z-m 3 (4) Xj. s= tt0 - U, + U a - Uj x 3 = u.- u,- tt^Lu, The permutations (01)(23), (02)(13), (03)(12) of x's in the group Q, send u 0, u, , u 2 p into u c, u,/-u i,-u 3j u 0, —u,, u^.-Ujj and u„, -u, , - u x , u 3, respectively. It follows that the basis functions (5) are unaltered by a l l the permutations of the x's in Q, Them's are easily seen to be algebraically independent since this is true of the x's and the u's. Finally, a rational function of the x's is a rational function of the u's, and is unaltered by a l l the permutations of Q i f and only i f the numerator and denominator consist of terms: t = c u„ u,' u z u 3 , whioh are themselves unaltered by Q. Under the permutations of Q,t beoomes, respectively: t, ( - 1 ) ^ ^ t, < - ! ) « ' a n d ( - l ) e ' ^ t . Henoe, i f t is unaltered by Qi (e^+ e 3),( < z-i +" -t-ej must a l l be even. Let e „ = ta> e, + e 3 = 2f, , e, •+ e 3 = 2^ , e, e^ = 2f 3 , where f„ , f, , f,, and f 3 are integers. Then evidently This oompletes the proof that J,, , «*,, and o<3 form a minimal basis for Q. T© express the elementary symmetric functions of the x's in terms of them's, we f i r s t determine: E, = u« E x = eu.o1"— 2(u,V U j \ u^ ) (6) E 3 = 4u„ — 4u a (u, +-u^  + u^- )-h8u, u^u^ B^«(u*t-u*+u*) ~2(u, 1- < + u, 1 ^ t u ^ U J ) + (functions of u«) But from equations (5): u ^ o/ a« ( u . u ^ - o<3 (7) uj- = «(, <*x Hence in terms of them's E x = (V. + ^. ^ 3 "t- ^  ^ J E 3 = 4-<*„ 3 - y ( X ^ +- ~ j +- ^  + (S) -^(powers of V 0 ) Consider now a reduced biquadratic equation x w+ a t xx+ a 3 x + a 4 = 0, (2) having the Quartio group Q with respect to R, and whose roots are xoa x, e x^, axx&3x , By the Galois theory, a function of the roots which is unaltered by, Q has a rational value. In partioular, the functions < o^s °4 , «*\, and o»3 have rational values, provided no single u^ is zero, i,3. Hence with this restriction on the u's equation (2) can be expressed in the form x*> a xx x4 a^x +- a,. = 0, a v = -2 C«f*. + "3) (9) a 5 = -8 ~\ ^3 a ^ = ( - + ^ H " J ~ z (< ^ + ^ "a", + % ^ < 0 where them's have values in R, Conversely, suppose rational values are assigned to the o£ 1 s in (9). Then, since the =<_8s are functions of the x's, form a basis for the group Q, and are rational, i t follows that every function belonging to Q has a rational value. Hence, by the Galois theory, the group of the equation in (9) is either Q or a subgroup of Q. The restrictions on there's whioh exclude the latter possibility are most easily found by considering the roots of (9). These are x 0 « V^T^s V*, c 3 x , = a T ^ V ~ (10) It is evident that no proper subgroup of Q is transitive; henoe equation (9) is reducible in R i f i t s group is a subgroup of Q. For example, i f i t s group is [ l , (0l)(23)J, i t is reducible into two quadratic factors such that x e -t- x,, xD x,, x^x,, xx x 3 are a l l rational. This means that 0 ( ^ 3 must be the square of some number in R. Similarly, — 4-- 8 -i f the group is [ l , (02)(13)j, V.^must be a square; i f [ i , (03)(12)] , ^.^a. must be a square; and i f ( f ) , < % , °i, *3 s o/,^ , must a l l be squares. In other words, i f the equation in (9) is to have the group C and not one of its subgroups, the produot of any two of the of's must not be the square of a number in R. We have ohosen a minimal basis which gives, i f not the simplest parametric representation of the coefficients of (2), at least a symmetric one. We have now to consider the possibility of the u's ("ii^j'Cs) vanishing. First i t is obvious that no two u's can be zero simultaneous-ly, for then (2) is reducible and cannot have the Quartic group. This is not the case, however, when only one of the u*s is zero. Suppose, for example, u z= 0. We then have a singular^ value for the parameter , which corresponds to a parametric representation for the group Q whioh is not included in (9). To cover this oase we must consider an alternative basis: e4'= u„ , u* , •: * (12) = u, 1*^=0, < = U* , S) which leads to the parametric representation: ' / f 2 k <*;>x +- (<' - «-3' f = 0 " (13) •where and range over a l l values in R for which oS, <*a is not a square. It i s dear that the cases where u, =• 0 or u s= 0 lead to the same form of equation (13). Moreover, equation (13) i s distinct from and supplementary to the equation in (9). We sum up the Quartic case then in 4) Of. Noether: Op. Git., p.229. 6) The procedure for deriving this formal representation and proving its validity i s similar to that for (9). Theorem 1; The equations x*- 2( s:^,^)x v- x-f ( 21o/,^_ g gL^-K-s)^ 0 (14) together include a l l biquadratic equations with the Quartic group Q when^r( <^3 ,*3 range over a l l values of R for whioh the product of any twomf' s, and the product of are not the squares of numbers in R. IV. Biquadratic Equations with an Ootio Group. A rational function of x,, x,, xz, x3> whioh is unaltered by the Ootio group 0 Q(13)J , is also unaltered by the Quartio group Q. But in addition i t is unaltered by the permutation (13) on the x* 8. Now such a function, as shown in Section III, is also a rational function of <*, 8^z. , % . Moreover, since the permutation (13) on the x* s merely interchanges «»< and < 3^, and does not alter °(. and «<, s a rational function of the x* s unaltered by 0 must be symmetrio in °<$ and^ 3 , when expressed in terms of them's. In other words i t is expressible as a rational funotion of the symmetric functions «t',^ «3 and «'.«,3 of < and<*j with coefficients whioh are rational functions of <=ia and^j. • Hence the funotions form a minimal basis for 0. This basis was chosen because i t was homogeneous; thus we hoped that i t would lead to a f a i r l y symmetric parametric representation of the coefficients of the reduced quart ic equation (2). The results: 10 -x*> a^x'-f a3x-^- a^ — 0 a ^ _ (a'aS -+ e,"*a») (16) a 3 = - * fa'a^") •whose derivation is similar to that of the case which follows, are, however, not general. For even when/?/', e/', /S^ take on a l l rational values (with certain restrictions to bar out subgroups ©f 0), the totality of biquadratic equations with the group 0 is not represented by equation (16), since the exceptional case, <=C,+«£=o, is automatically excluded when we employ the basis (15). Moreover (2) may have the Ootie group 0 when V, -*"^  = 0. Hence we use the functions; <3o = *° y /3, <= <*, •+ ~>3 f (17) whioh by the same argument as that for (15), form a minimal basis for the group 0. How suppose that x„, x,, x 2, and x, are the roots of a reduoed biquadratic equation having the group 0 with respeot to R. The functions B in (17) are unaltered by 0, and henoe have rational values, provided, as in Sectioning no single u,- is zero. After constructing the elementary symmetric functions in terms of the & ' s, we see that, when no u is zero by i t s e l f , the equation having the Octic group with respeot to R has the form: x.*-4 \ x2+ a3x-*- a* = 0 a z = - z(/3, /3,_-t/33) (18) a 3 s - g /33 - 11 -where the ( s are rational. Conversely, an equation (15) with the & * s rational, has a Galois group with respect to E which is either 0 or a subgroup of 0. Wow the proper subgroups of 0 whioh we need to consider are the Quartio, the Cyclic, and the group K2 [ l , (02), (13), (02)/ (13)1 ; and subgroups of these. If the Galois group of (15) is the Quartic group Q or one of i t s subgroups, the functions x„x, + x Ax 3 , x / x 3 and x8 x •+ x x i , whioh are incidentally the roots of the resolvent cubio of (15), must a l l be rational. In terms of the a 1s, this means that 4S3 must be the square of a number in E. If the group of (15) is the Cyolic group C. or a subgroup of i t , the function x^x,+ xfx^-y-x**, + x^xo must be rational; and i f the group K, xa-t-xz— x,- x 3 must be rational. The conditions oft the 6 *s may be summed up by saying that, i f equation (18) is to have the Octio group 0 and not one of its prep*** subgroups, then /S, - V<?3 , &3 must not be the squares of numbers in E. We must now consider the singular values of the parameters which arise when any u vanishes. There is obviously no diff i c u l t y when u,= 0, or u } = 0, or any two of the u's equal zero simultaneously, for then the group of (18) is a subgroup of the Ootic. Let us suppose then that u x alone vanishes. Employing the s defined in (12) we construct the basis functions: It i s easily verified that the a''s form a rational minimal basis for 0 whioh gives the parametric representation: - 12 -x^-2^'x%( a,f - ^ 4 ' ) = 0 (20) where ./st ' and/&J have rational values. To ensure that (20) has not a f»-°p<t>~ subgroup of 0 for its Galois group, /s/'—^/s^' and/?3' must not be squares. To complete the Ootio case then, we oombine (18) and (20) in Theorem 2; The equations * H~+-± V f*;*- «&3>) = (2I) together include a l l biquadratic equations with the Ootio group 0, when at /5i BHA/33 ; <3,\A3' range over a l l values of R for whioh ¥/3a / <&3 ^  , 4 and s'*_iia' gj are not the squares of numbers in R. NOTE; It w i l l be observed that the pure equation x*y- a^ = 0 (22) has the Ootio group when a v is not a square , and the Quartic group when a ¥ is a square,, V» Biquadratic Equations with the Alternating Group. As in the case of the Ootic group, a rational function of the independent variables xa>xtyxl} x3 which is unaltered by the Alternating group, i s also unaltered by the Quartic group. In addition, however, i t is unaltered by the permutations (123), (132) on the x's. Moreover, this rational function of the x's is also a rational function of the o(. } s ; and since the cyclic permutations (123), (132) on the x's effect the same permutations on the << 's, i t follows that any rational function of the x's which is unaltered by the Alternating group may be 6) Of. Seidelmann, F.; Op. Git., p. 232. - 13 -expressed as a rational oyolic function of andc^ with coefficients which are rational functions of U0 . We find i t convenient to introduce the irrationality ^> , where ^  is a primitive cube root of unity. Let y„ = °sa 3y, = <*\ -f ^ (22^ 3y r = *s, -t uj c 3 ie/'-cJ^  3y 3 = ^ ui " j Solving for the ^ ' s -. - y. * ys.v-y3 y ^ ' y ^ y , (23) From what has been said above, i t is dear that f » - Jo* y x = yz y 3 i, y,» y 3 =- yt+y/ (24) forms a rational minimal basis for the group A. In terms of the Y ' s, the elementary symmetric functions of the x's turn out to be ! i = - ^ (.*,*- M E 3 = *(*,*-*-*3 - 3*< V*) (26) Now consider x ^ x ^ x^ and x 3 as the roots of a reduced biquadratic equation having the group A with respect to B. The functions y in (24) are unaltered by A, and hence have rational values. We need make no provision here for any of the u's vanishing, for, i f one of them vanishes i t can be shown that equation (2) reduced to the form x + a z x -f- a^ .= 0 whioh has either the QuartiG or Ootio group; and i f two or more of (26) u/* uz o r u 3 vanish there is a s t i l l further reduction of the group. Hence an equation with the Alternating group A with respeot to R has the form x^V \ x1+ a^  x +• a ¥ = 0 * ^ = - &(r,2-a 3 = -r(Tr,3-+ys-3Y,^) =• -s fa ¥- * x, y 3 + 3 ^ 3 y, a * i ) and conversely, an equation(26) with r a t i o n a l y 1 s has either the group A or a subgroup of A. The possible subgroups of A which (26) may have are the Quartic group, and those four groups which are oyclic on three of the roots of (26). If the group is the Quartie, the roots of the resolvent oubio of (26) must a l l be rational; i f the group is one of the four groups whioh are oyclio on three of the roots, the other root must be rational and equation (26) reducible. We are now ready to state Theorem 3: The equation C(Y,z-K)-X*-- *(r, fjn-sfc ti/*,r3 3 Y, ° (27) includes a l l biquadratic equations whioh have the alternating group A; Xl} Y, , Yz may take on a l l values in R for whioh (27) is irreduoible, and for which the resolvent oubio equation of (27): y 3 — a^y 1— 4a ¥ y + ( i a j i * - a 3 z ) = 0 (28) with a J } a 3 a ^ defined as in (26), has not a l l three roots rational. As a point of interest we mention a perhaps more practical criterion for fourth degree equations with the alternating group. If the Galois group of equation (2) is the alternating group A for the f i e l d R, then the square root of the discriminant of (2), namely, P = (x,,- x, ) ( x a - x ^ X x . - x ) ( x r - x ) ( x , - x*)(x^ x 3) - 15 -must have a rational value; and conversely, i f P has a rational value, i t is unchanged by the Galois group of (2), whioh must thus be the Alternating group A or one of i t s subgroups. Hence Theorem 4: The equation x*+ a^  x* + a,x + a^ . = 0 (2) embraceseall biquadratic equations with the Alternating group when the a'8 take on a l l values in R for whioh: (a) the discriminant A = 4 (4a a J- )3-z7 (%\ a ^ — af )*" is equal to the square of a number in R, (b) equation (2) has no rational root, and (o) the resolvent cubio has not a l l its roots rational. VI. Biquadratic Equations with a Cyclic Group. In order to oonstruct a minimal basis for the cyclic group C = [ I, (0123), (02)(13), (0321)J , i t is convenient to introduce, temporarily, the irrationality,!, where i = ^-l. Let 4z„ = x e x, f x t -f- x 3 4z, = x„ -f- ix, - x x — i x 3 4z z = x 0 — s, + I,. - x 3 4z = x„ - ix, — x^-f- i x 3 Then x,, = z D •+ g, +- z z z 3 x / ; — z e — iz, - z ^  + i z 3 X t = Z. - Z, E a - Z 3 x 3 =. z 0 •+• i z , - z ^ - i z a (29) (30) The permutation S= (0123), which generates C ? oarries z e, z, , z^, and - 16 -into z a , - iz, , - z l , and i z 3 , respectively. By means of the z's, i t is easy to construct funotions of the x's whioh are unaltered by S, and hence by C; for example, the funotions; A l l of these functions of the x'8 have coefficients in R ( i ) , but not a l l .ooeffioients are in R. We f i r s t find a minimal basis for the rational functions $ (x) of the x's, unaltered by C, with coefficients in R ( i ) ; and subsequently, find a minimal basis consisting of functions with ooeffioients i n R. It is clear, by (2), that a function <]) (x) is expressible as a rational function of the z's with numerator and denominator consisting of terms of the form t = cz 0 z,' z 3 which are unaltered by C. The permutations S —(0123), S 2, S 3 send t into (-1) ^ ( i ) * ' ^ t, (- i ) * ' + ^ t , and ( - l f ^ i ) * ' " " * 5 t; respectively. Since t is unaltered by C, we must have(e,+ es) and thus {&,— ©j),and (e j.£ e,~ 03 ) even» Let ©, •+- e 3 — 2f £ , © z + e,- e, -= 2f , e i- e,- e> _ 2f^ , Then t = o •/* (z z3 > (z* z^) f' (g* z j f * It follows that the functions & = «„»• = *, *3, ft - P^z^z^ (31) which are themselves unaltered by C, and are readily seen to be algebraioally independent since the z's are, form a minimal basis for the funotions (j) (x). It is now not diffioult to construct, by means of the (p 's, the required minimal basis, with rational coefficients, for the group C« F i r s t , i t is clear that the functions : -17-(32) also form a minimal basis for the functions (^(x). Moreover, the o"s as funotions of the x's, have ooeffioients in R, This is easily verified for cG^ty* and o£ -<£| , and is true for cf, and <^ since the replacement of i by - i in the ooeffioients of the x's interchanges z, and z ? , leaves unaltered, and hence interchanges ^ and , Let f(x) be a rational function of the x 1s with rational ooeffioients. Then f(x) is also a $ (x), and henoe is a rational function of the <p *s with coefficients in R(i). Such a function of the (p ' s evidently has rational ooeffioients, as a function of the x* s, i f and only i f i t is unaltered by the replacement of i by - i . Let t = d t« <?, ' ^ j T j 3 be a term of the numerator or denominator of f(x), and let t be the expression obtained from t by the replacement of i by - i . Then ¥ ss. d* Q>*°(p,3t'<?,9*@f', where 1 is the ooajugate of d in R(i). If t has rational coefficients, as a function of the x5 s, t=T; whence d="d, g r g,* Thus t = dfa,<?s)K j[<S?*<£9*(<r?*+4?J3. which is a rational function of the QP*a with rational coefficients* If t 4= T, then the numerator, or denominator, of f(x) must contain t when i t contains t; that i s , i t contains t + T = fi*%*+Z%*?s9') It is a simple matter to show that the expression in parenthesis is a poly-nomial in and with rational coefficients. This completes the proof that the <P%s form a rational minimal basis for C, sinoe f(x) is thus expressible as a rational function of the </**s. In terms of the z's, the elementary symmetric funotions of the x* s are: . •. - 18 -E^= 6z0* — Zz£ — 4z, z3 E 3 = tzj - 4 z 0 ( z z V 2 z , z3) + 4z^(z,Vz/) (33) E* = zt ~ 2 z / ( z ^ 2 z / z ^ ) ^ 4 z 0 s j z / + s 3") Thus E 2 E f + (34) Now let x , x , x , x be the roots of the reduoed biquadratic equation x V a , x V a , x + a ^ = 0, (2) whose group with respeot to R is C. It is clear that the c/"' s are functions of the roots, whioh are unchanged by the permutations of C and therefore have rational values. Hence the coefficients of (2) can be expressed in the form *3 = ~ f ( (35) where the cT% 3 have values in R. As in the case of the Quartio group, it is readily shown that, conversely, i f rational values are assigned to them's in (3$), the group of equation (2) is either 0 or one of its subgroups. The only proper subgroup of C whioh we need to consider is the intransitive group £ I, (02)(13)J . If (2) is to have this group, then i t must be reducible into two quadratic factors such that x -r x , x x . x +• x , x x are a l l rational. This leads to a restriction on the <f*B whioh is included in Theorem 4: The equation ranges over a l l fourth degree equations with the cyclic group C when '» •» °3 a r e assigned a l l values in R such that / tf2+ not the square of a number in R. -20 -1. Seidelmann, F: 2. Noether, E: 3. Breuer, S: 4. Dickson, L.E: 5. Carmiohael, R.D; BIBLIOGRAPHY "Die Gesaratheit der kubisohen und biquadratisohen Gleichungen mit Affekt bei beliebigen Rationalit&tsbereioh. a Mathematisohe Annalen, 1918, pp. 230-233. "Gleiohungen mit Vorgesohriebener Gruppen" Mathematisohe Annalen, 1918, pp. 221-229. "Mctasyklisohe Minimalbasis und complexe Primzahleri" Journal fur Mathematik, bd. 155, 1927, pp. 13-42. "Modern Algebraic.Theories." Benj. Sanborne & Co., New York, 1930. "introduction to Theory of Groups of Finite Order. Ginn and Co., New York, 1937. 


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