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On the Galois groups of certain algebraic number fields Straight, Byron William 1949

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ON THE GALOIS GROUPS OP CERTAIN ALGEBRAIC NUMBER FIELDS by BYRON WILLIAM STRAIGHT A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN THE DEPARTMENT of MATHEMATICS THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1949 Abstract of Masters' Thesis On the Galois Groups of Certain Algebraic Number Fields by Byron William Straight April, 1949 This thesis i s concerned with the Galois groups of the root fields of the equations xP - a = 0, (x p - a)«(xq - b) = 0 and (x q - b ) p - a » 0, where p and q are distinct primes, and a and b are rationals. The correspondence of subflelds and subgroups i s studied for each of the three cases. The f i e l d F(o<., d£) formed by adjoining to the rational f i e l d F the elements andoc, a primitive pth root of unity, i s shown to be the root f i e l d of x p - a = 0, normal over F of degree p(p-l). The Galois group of F(o£, 5£T) over F Is found to be the metacyclic group constructed from generators s and t subject to relations s p = 1, t p 1 = 1 and st = t s r , where r i s a primitive root modulo p, and where s is the automorphism which maps Ba~ onto oL-^/a. while t i s the automorphism which maps c*. onto oCr. Various subgroups and corresponding subflelds are studied and nine theorems proven on their correspondences, illustrated with a partial lattice diagram. The f i e l d F ( ^ , ^ , % ) where (3 Is a primitive qth root of unity i s shown to be the root f i e l d of (x p - a)-(xq - b) = 0 and the Galois group i s proven to be the direct product of two of the %ype for the f i e l d P(oC , $T). The f i e l d P(J5, St + oC1- $0 for I - 1, 2, 3 ... p, which i s the root f i e l d of the equation (x q - b ) p - a = 0 i s studied and shown to have degree pqp-(p-l)*(q-l) • The Galois group i s found to be generated by four independent generators: s, t, u, v subject to eleven defining relations. Here the elements s, t, u, v are the auto-morphisms which respectively map 3a~ ontoe/v- c?C onto c<T, S b~7^ onto P'% + d& and j3 onto f$W, where w is a primitive root modulo q. A partial lattice diagram illustrates the correspondence of subgroups and subflelds. The thesis was carried out under the supervision of Dr. D. C. Murdoch. CONTENTS Chapter One I n t r o d u c t i o n , D e f i n i t i o n s , Fundamental Theorems, and Statement of the Problems Chapter Two The A l g e b r a i c Number F i e l d F(oG •ya) Chapter Three The G a l o i s Group of Ffc£, $ 6 " ) over F Chapter Four The Subgroups of the G a l o i s Group Chapter F i v e The G a l o i s Correspondence of Subgroups and S u b f l e l d s Chapter S i x The G a l o i s Group of Fj$=t, yb), and the G a l o i s Correspondence of Subgroups and S u b f l e l d s Chapter Seven The G a l o i s Group of , % + ©c1 $a~) and the G a l o i s Correspondence of Subgroups and S u b f l e l d s B i b l i o g r a p h y - 1 -CHAPTER ONE Introduction This thesis is concerned with the Galois groups of certain fields and with the correspondences•of the subgroups and subfields. The algebraic number fields to be considered are the following three (where p and q are distinct primes). a. F(e*, ?/£) obtained by adjoining in the usual sense to the f i e l d F of rationals a l l the roots of the equation x P - a = 0. Here<^ Is a primitive pth root of unity, and a i s a rational which i s not a perfect pth power of a rational. b. F(«<,^> obtained by adjoining to the f i e l d F of rationals a l l the roots of the equation (x P - a)-(xq - b) = 0. Hereof i s a primitive pth root of unity,^ i s a primitive qth root of unity, a i s a rational which i s not a perfect pth power of a rational and b i s a rational which i s not a perfect qth power of a rational. c. F(p, % +oC1-9£) for i » 1, 2, 3 ... p, obtained by adjoining to the f i e l d F of rationals a l l the roots of the equation (x p - a ) q - b = 0. Here a, b are rational indeterminate3,0^ is a primitive pth root of unity and ^  i s a primitive qth root of unity. The discussion depends on several well known definitions and theorems of the Galois theory* which we .* See for example ( l ) , (2), (3). -2-shall now state. (a) If the elements , u g , u 3 ... u m form a basis over P for a f i n i t e extension K of P, while w , w , w ... w 1 2 3 n form a basis over K for an extension L of K then the m*n products u ^ j for i = 1, 2, 3 ... m and J • 1, 2, 3 ... n form a basis for L over P. (b) If K is a f i n i t e extension of P and i f L i s a f i n i t e extension of K then L i s a f i n i t e extension of F and i t s degree {LSF] - iLrK] • (k:F]. (c) A normal f i e l d N over F is one in which every polynomial Irreducible over F which has one root In N has a l l roots in N. (d) A f i n i t e extension of F i s normal over P i f and only i f It is the root f i e l d of some polynomial over P. (e) The Fundamental Theorem of the Galois theory: If K i s a normal algebraic extension of a f i e l d F and G the Galois group of K over F consisting of those automorphisms of K which leave fixed a l l the elements of F, then there i s a one to one correspondence of subflelds of K and subgroups of G such that: 1. to each subgroup H of G corresponds a f i e l d L of K consisting of those elements l e f t unchanged by the automorphisms belonging to Hj 2. to each subfield L of K where L D P there corresponds a subgroup H comprising a l l -3-elements of G which leave each element of L fixed! 3. the subfield L i s normal over P i f , and only i f , the subgroup H i s normal in G, (in which case the automorphisms of L which leave P fixed elementwise form a group isomorphic to the factor group G/H^  4. for each intermediate f i e l d L the degree of L over P Is equal to the Index of H in G, and the degree of K over L is equal to the order of the Galois group H of Lj 5. conjugate subfields correspond to conjugate subgroups j "6. the union of two subfields and Lg corresponds to the intersection of the corresponding Galois subgroups and Hg* and dually r\ L g corresponds to v Hg. (f) A polynomial ^  - a of prime degree over the rational number f i e l d i s either Irreducible over the rational number f i e l d or has a root In the rational number f i e l d . (g) Every element w of a f i n i t e extension K of F i s algebraic over F and satisfies an equation irreducible over. F of degree at most n where n => [k:F3 . -4-CHAPTER TWO The Algebraic Number Field Ffel, Let F be the rational number f i e l d , a be a rational which is not a perfect pth power of a rational, p be a prime, and o(. a primitive pth root of unity. Theorem 2.1 The f i e l d FfBSj^has over F,degree p. Proof: Since a i s not a pth power of a rational then x p - a - 0 has no root in F and by theorem (f) of the introductory chapter the equation is irreducible over F. Then by theorem (g) the f i e l d F(?/a~) has over F a degree not less than p. From the defining equation F($£) has degree«over F, not more than p. Theorem 2.2 If ©<. and are adjoined to F, the resulting f i e l d is normal over F. Proof: Since < satisfies x*5""1 + x P~ 2 + ... + x + 1 « 0 and satisfies x p - a = 0 and since a l l their conjugates are obviously present in F ( o ( , F ^ ) , the result follows from theorem (d) of the introductory chapter. Theorem 2.5 The f i e l d F ( e O has,over F,degree (p-1), and the f i e l d F(«»C has,over F,degree p(p-l). Proofs Since satisfies the cyclotomic equation xP" 1 + x p" 2 + ... + x + 1 = 0 which Is irreducible over F*, then l , ^ , o < . ...,<p" form a basis for F & J over F ?of degree p-1. Since F(«<, Is a s p l i t t i n g f i e l d for x p - a • 0,where a i s in F and hence in F(c?0, * See page 102 of (2) -5-i t follows that F(o(, f/S) is a normal extension of F(o() and cyclic over F(o0 with degree p over F(<?0.* From theorem (b) of the introduction i t follows that F(c\, tya) has,over F, degree p(p-l). In order to determine subfields of K *= F(s\, i t is instructive to study the Galois group of K over F. * See page 61 of (1) -6-CHAPTER THREE The G a l o i s Group of F(ol, ?/a") over F By theorem (e) of the I n t r o d u c t i o n the order of the G a l o i s group of K = F(e<, ^a*),over F , l s p ( p - l ) , t h e degree of K over F. Theorem 5.1 The G a l o i s group of K over F i s generated by the automorphisms s, t , xvhere : under s, *• l e a v i n g c< and F i n v a r i a n t , under t , «< * e*?*, l e a v i n g ?/S and F i n v a r i a n t , where r i s a p r i m i t i v e root modulo p. The elements s, t are subject to the ( s o l e ) d e f i n i n g r e l a t i o n s s p = t = 1 and s t = t s r j they generate the meta c y c l i c group of order p ( p - l ) . Proof: A b a s i s f o r K over F c o n s i s t s of the p ( p - l ) 2 p-2 products of the b a s i s elements 1, , <K. ...<*. and To observe the e f f e c t on any number i n the f i e l d K,of an automorphism of K l e a v i n g F unchanged elementwise, i t i s only necessary to observe the changes i n the b a s i s elements, which are j u s t powers and products ofoi. and Since oC s a t i s f i e s m(x) e x p - 1 + x p ~ 2 ... + x+1 = 0, any automorphism l e a v i n g F I n v a r i a n t , (and hence the c o e f f i c i e n t s of m(x)) must mapoC onto another r o o t of the same equation; i . e . , onto another power of oC . S i m i l a r l y tya must be mapped onto another root of x p - a = 0. Hence the only p o s s i b l e automorphisms are a l l those generated -7-by s and t . Under s, % w under s 2 , % *-d?^t p - l p/-under s , ya and under s p , ?/S Obviously the order of s Is p. Under t, p r i m i t i v e root modulo.p, under t 2 , *~(o<F)r =0^, 3 3 under t , »o^" , *- <*T where r is a under t , . K • To be sure that the powers of t map oc onto a l l p o s s i b l e conjugates we must choose r to be a p r i m i t i v e root modulo p. r m I f m i s . the order of t i t Is necessary th a t = ocbe such that r belongs to p - l mod p, which i s s a t i s f i e d i f r i s a p r i m i t i v e root modulo p. Then t has order ( p - l ) . Under s t : % *- dftfa, o{ *- o{ >- o^1* - and under t s r : Ja" — ' »- %. ~ *-*- o^r >- 6 < R . Then s t = t s r . The exi s t e n c e of the r e l a t i o n s s p = t p 1 « 1. and s t = t s r has now been proven. I t has been shown that a l l the elements of the G a l o i s group of K over F are generated by s and t . I t i s s t i l l necessary to show tha t the r e l a t i o n s above are complete; that i s , t h a t they are r e s t r i c t i v e enough to l i m i t the order of the group generated by s and t to p ( p - l ) . Consider a l l the elements of the form t c s d 9 where c » 1, 2, 3, . . . p - l and d: =* 1, 2, 3, ... p. These are a l l d i s t i n c t , f o r i f t c s d = t g s h (reduced) then t G " g = s h ~ d . But because of the nature of s and t t h i s i s impossible unless each i s the i d e n t i t y , whence c = g and d = h. By repeated use of the r e l a t i o n s t ~ t s r (which I s equivalent to the more gen e r a l r e l a t i o n s m t = t s m r ) every element generated by s and t c d can be w r i t t e n i n the form t s . Then the group has order p ( p - l ) . The r e l a t i o n s e s t a b l i s h the G a l o i s group. CHAPTER FCUR The Subgroups of G the G a l o i s Group of K over F Theorem 4.1 The subgroup generated by s, written {s} , i s normal i n the whole group G. Proof; Transform the subgroup {s} by any element s K tf* of G. (In the previous chapter,, any element of G was shown to be representable In t h i s form). The r e l a t i o n s t = t s r can be extended (by repeated use) to s m t * = t* s m r \ Therefore, t V - {s} . s*t> - t * . [s] • * = t x . * • {sj r* = {s} . Theorem 4.2 A l l the Sylow subgroups of G are c y c l i c . Proof 0. The order of G i s p ( p - l ) . Since {s} i s normal the only Sylow subgroup of order p i s [s] which of course i s cy c l i c . . Let q r a be the highest power of the prime q which d i v i d e s ( p - l ) . A Sylow subgroup of th a t order I s generated, by (t)*' • I t i s obv i o u s l y c y c l i c . Hence a l l the Sylow subgroups of that order are c y c l i c , being conjugate.* Theorem 4.5 The group G contains a t l e a s t one d i h e d r a l subgroup. Proofs A d i h e d r a l group i s one generated by two elements 2 2 A and B subject to the ( s o l e ) r e l a t i o n s A = 1 = B and ( A B ) m « 1. A d i h e d r a l subgroup can be e x h i b i t e d by •p-i choosing f o r A the element ( t 8 ) and f o r B the element ( t s ) 2 . Obviously A = 1. Now using repeatedly the r e l a t i o n s t = t s r , we have * See p. 58 of (3) - l O -fts) 2 - t S - t S - t 2 S r + 1 5 (ts) 3 - t S . ( t V + 1 ) = t V - t S ^ 1 - t V 2 + 7 + 1 » ( t s ) n - t ^ . s ^ " 1 + ^ + + and and ( t s ) p _ 1 . ( t P " 1 ) . s : JP-2 + rP-3 + _ + r + i 2 P-2 But r Is a p r i m i t i v e r o o t modulo p, hence 1, r , r ... i r are congruent i n some order to the numbers 1, 2, 3 . . . ( p - l j . These t o t a l P^'1K Hence ( t s j P - 1 - tP"* 1 = s ^ f " 1 ^ = 1. A l s o , (p-1) i s e x a c t l y the order of ( t s ) , f o r any power l e s s than p-1 would produce ( t s ) m = t m s n w i t h m^O modulo (p-1). Hence ( t s ) can have order no l e s s than p-1. Then ( t s j ^ chosen as element B has order 2. In the f i n i t e group 6, the t h i r d r e l a t i o n ( A B ) m = 1 i s of course s a t i s f i e d . As a matter of i n t e r e s t the value of m can be determined. I n - t h i s case, ( A B ) m i s j u s t f p - i , pp-i P-l rD m [t"^-(ts)« J = [ta •t"T--s (2| m f o r some value d. Hence / n_ xm d.m (AB; = (s ) = 1 and m « p. F i n a l l y the order of the d i h e d r a l group i s 2p.* Some w e l l known r e s u l t s on the m e t a c y c l i c group of order p ( p - l ) may be s t a t e d f o r the sake of completeness:** a. The only element which commutes wi t h a l l other elements i n the group i s the i d e n t i t y . , b. A l l i t s automorphisms are Inner automorphisms. c. The group i s isomorphic t o a doubly t r a n s i t i v e permutation group of degree p. * See pp 181, 182 of (3) ** See pp 123 - 126 of (4) and pp 181 - 184 of ( 3 ) . -11-d. In the representation of (c) the group contains a subgroup of even permutations known as the half-metacyclic group. e. The group i s the largest transitive group of degree p that contains a cyclic normal subgroup of degree and order p. f. It Is the largest transitive group of degree p which Is solvable. -12-CHAPTER FIVE The Galois Correspondence of Subgroups and Subflelds A p a r t i a l l a t t i c e diagram of subflelds and subgroups i s shown on the next page. The diagram was constructed from knowledge of the subgroups of the Galois group and by the use of part ( 4 ) of the Fundamental Theorem stated i n the introduction. The l a t t e r can be i l l u s t r a t e d by the p a i r i n g o f f of chains of subflelds and subgroups i n the following manner: f i e l d s D L ^ L 3 F 1 2 groups I C ^ c H g c G The f i e l d L i s the set of a l l numbers i n K which are l e f t invariant by the automorphism group EL^ . Since G has order p(p-l) then H 1 has order —jjj?--^-* and hence has,over F, degree m l Theorem 5.1 Corresponding to the whole f i e l d K i s the i d e n t i t y automorphism and corresponding to the ground f i e l d F i s the Galois group G. Proof; This i s t r i v i a l i n view of part ( 4 ) of the Fundamental Theorem of the introduction. Theorem 5.2 Corresponding to the group {s} of order p i s the f i e l d F & ) of degree (p-l) over F. Proof: Since {s} has order p, the f i e l d corresponding (the largest s u b f i e l d of K l e f t invariant by a l l of {s} ) } must have degree p - l . Under s , P/a" »- o^^ei, while F and and a l l i t s powers are l e f t invariant. Then F(o{) P A R T I A L L A T T I C E O F T H E F I E L D • F ^ 5 6 ) A N D I T S G A L O I S G R O U P O V E R F . K E Y S U B G R O U P GENERATORS CORRESPONDING SUBPIELD / ^^^^^^^^^ conjuofaT-es — c o n j u g a t e s c o n j u g a t e s -13-i s c e r t a i n l y contained i n the l a r g e s t s u b f i e l d l e f t i n v a r i a n t by a l l of {s] . But P(oC) i t s e l f has degree p-1, and {s}, having order p, could not correspond t o a f i e l d of degree g r e a t e r than p-1. Then F(e<) corresponds to {s} . Theorem 5.3 Corresponding to j t ] of order ( p - l ) i s the f i e l d F(tya) of degree p over F. Proof: We seek a f i e l d of degree p. Under t , < < — , while F and ?/a" are l e f t i n v a r i a n t . Now F($a") has degree p over F. Hence F(?/£) i s not merely contained i n the f i e l d l e f t I n v a r i a n t by jt] , i t a c t u a l l y i s the f i e l d l e f t i n v a r i a n t by [ t ] . Theorem 5.4 Corresponding to { s n t s - n } of order (p-1) i s the f i e l d F(d^"n- P/a) of degree p over F. Proof: From theorem 5.3, {t},of order ( p - l ) v l e a v e s F(^a") i n v a r i a n t . By p a r t 5 of the Fundamental Theorem i n the i n t r o d u c t o r y chapter, conjugate f i e l d s correspond t o conjugate groups. Hence the s u b f i e l d s F , ... F(peP~^'J%[) must correspond i n some order to the subgroups [sts""^, j s 2 t s 2 ] , ... j s p ~ ^ t s ] . Then the s u b f i e l d s must have the same degree over F as t h e i r conjugate F($T), which i s p. The exact correspondence i s e a s i l y found. From the r e l a t i o n s t = t s r we f i n d s m t s " m = t s m ( r ~ 1 ) . Under s m t s " m , oL~m(pJk~) > o C ( p - m ) r + m(r-l) Hence { s r a t S - m } leaves F(cK P~m-^a) i n v a r i a n t . In p a r t i c u l a r { s t s " 1 ] = [t s 1 " " 1 ] leaves i n v a r i a n t FO*" 1 - ^ ) while { s ^ t s } leaves I n v a r i a n t F(ctjtya), which are d i s p l a y e d i n the l a t t i c e diagram. -14-gheorem 5.5 £ h e g r o U p generated by (ts) corresponds to the f ield F(c*_n- P/a") where n Is any integer satisfying n = ^r'lj f o r k P 3 o m e multiple of p. Proofs As shown in the proof of theorem 4.3, the order of (ts) is (p-l) . The f ie ld left invariant under (ts) must have,over,F degree p. Under (ts), ?/a~ *bJ&> and *- oC r, whence otn- tya. ^ o c r n + 1 - H e n c e J?- fya is invariant under [ts] provided n is1 chosen to satisfy kp-1 n s rn + 1 mod p j that i s , n » —^rzj; for n an integer, with kp some multiple of p. This f ield F(e*.np/£) is one of the conjugates mentioned in theorem 5.4, and hence {ts} is a transform of t. Theorem 5.6 The group generated by (st) corresponds to the f ie ld F(oCm,P/a") where m is anyinteger satisfying kp-r m = rTTX for kp some multiple of p. The proof is omitted, being very similar to that of theorem 5.5. Again the field is a conjugate of F($£) mentioned in theorem 5.4. Theorem 5.7 The field corresponding to the group generated by t* 1 is L = F(P/£, &m) of degree np, where nm = p - l and m 2m (n-l)m e m e o t + ^ + < K + + <* Proof; Consider the set of SL^ and its transforms by the automorphism t and its powers. 2m (n-l)m + . . . + oC 2m+l (n-l)m+l + • « • + o<. . . 2m+2 (n-l)m+2 r + ...+<*. 9*= + m+1 r' + cC X + X r 3n= oC + X =15-O • 0 » •••• o o • * tm-l m-1 r3m-l rmn-l The powers of oc o c c u r r i n g i n these, form a complete s e t of d i s t i n c t powers of r from 0 to (mn - 1) , or (p - 1) i n a l l . No two. of these transforms of & mcan be equal, f o r the equating of any two of them would g i v e r i s e t o an equation i n o c _ o f degree l e s s than (p - 1) w i t h r a t i o n a l c o e f f i c i e n t s . T h i s i s impossible s i n c e o C i s a r o o t of the cyclotomic equation + x p ~ 2 + ... + x + 1 - 0 which i s i r r e d u c i b l e over P. Now consider the f i e l d P(^a", Under s, —*-*tyk~i hence no power of s can belong to the G a l o i s group of t h i s f i e l d . Under t* Qm i s u n a l t e r e d f o r A- any m u l t i p l e of m, f o r the powers of «< o c c u r r i n g i n Om w i l l then merely be permuted among themselves. I f A i s not a m u l t i p l e of m then ^ , i s transformed Into ^ t * $ t h a t i s , the va r i o u s transforms of ^ w i l l be permuted. Then F($a~, i s i n v a r i a n t only under the group where m = ^ pp. As an example, s p e c i f i e d i n the l a t t i c e diagram, we have F(P/£~, [<=L+ corresponding t o { ^ } Theorem 5.8 Corresponding t o the group generated by s and t m , where m-n - (p-1), i s the f i e l d F(Gm) of degree m over F, where © =c<+oCr + ... + o^r 'mi ~m Proof? There have been e s t a b l i s h e d the correspondences {s} , order p ( ; FfcO > degree (p-1), - 1 6 -and {t m} , order n ^ ) , degree, mp , where 0 i s the element defined in theorem 5 . 7 . Under m unions and cross-cuts by part (62. of the Fundamental Theorem in the introduction we Immediately have the correspondence : {s} o {t m},order np F(P/a~, n F(<=£),degree m The group i s just the group (s, t m ) of order P m^*"'^  = n p * because any element of the form s°-(t m) d can be written (t m f - s e for some value e (by the relation of theorem 4 . 1 ) . Then the union of {s} and {tm} comprises a l l elements of that form. Hence the f i e l d corresponding must have.over F, degree ^np 1^" - m* Since F(fya~) i s not contained in the f i e l d F(oC) then It i s only necessary to state that the cross-cut on the right i s the f i e l d F(^„) completely contained in F ( o C ) . Theorem 5 . 9 The group generated by (st m) corresponds to the f i e l d F(oC n •?/£", Qm), where n = k P * ^ for^an integer and kp some multiple of p, and where e m i s the element defined in theorem 5 . 7 . Proof: It i s required to show that s t m i s a transform by s n of the element t», for a value n to be determined. Assume s " n t s n = s t m . T,hen t m s n - an+1-tm = t m ' S ( n + l ) r l m from the relation of theorem 4 . 1 . Hence n = (n+l)rm,mod kp+r13 p, or n i s an integer satisfying n = ^ ^ for kp some multiple of p. For this value of n determined from the , , -n m n m given values of p, m, r we have s t s = st „ The group {stm} i s a transform by s n of the group {tm} and hence Its -17-corresponding f i e l d i s a transform of F(?£, established in theorem 5.7. It i s obvious then that (stra} w i l l correspond to F(<<n-5a, e ) where n = l c p" i" r m as stated above, m 1-r since the transformation by s n merely changes F ( ? / & ) into F(e<n-^a") corresponding to [s n t s n ) . Decreasing the order of the group from p - l to p - l increases the degree of the m corresponding f i e l d from p to rap. The analysis of this correspondence could.be continued by considering conjugate gubgroups of those given In the previous theorems, and by investigating cross-cuts and unions. But we w i l l turn now to a different equation, f i e l d and group. • -18-CHAPTER SIX The Galois Group of F U ^ , Fya", % ) Consider the equation (xp - a)«(xq - b) = 0 where p and q are distinct primes, and a and b are not (respectively) exact pth and qth powers of rationals. Adjoin a l l the roots of the equation to the rational f ie ld F. This is equivalent to adjoining the numbersoC,/^, $/at IJfo", where ©c is a primitive pth root of unity and @ is a primitive qth root of unity. Theorem 6.1 The f ie ld F (©< . ,^ , P/S, %) is normal over F. Proof: The conjugates of a l l the basis elements (over F) are present, hence the conjugates of any number in the f ie ld are present, and so the f ield is identical to its conjugates. Theorem 6.2 The Galois group of the f ield F(oC ,@ , ffi) is the direct product of the Galois groups of F( oL , ¥fa) and F(/3 , S ) . Proof: Both F(o<_ , -5a~) and F( @ , $b) are normal over F, hence correspond to normal subgroups of G, the whole Galois group, by part (3) of the Fundamental Theorem. Let L^ be F(oC, and Lg be FQf?, %) and let and Hg be the corresponding normal subgroups of G. Now L^u Lg = F{pL,@, P/a~,-S^j hence H g = 1. Since F(c<, 8a~) and F ( ^ , 3b~) have nothing in common except the ground f ie ld F, then "L^r\ L 2 = F. Hence H g = G. From H ^ H g = 1* YL^yE^ • G, and the normality of and Hg in i t follows that G = % x Hg. The l a t t i c e d i a g r a m f o r G,or F ( « C , / 9 , i s o b t a i n e d by f o r m i n g a l l p o s s i b l e u n i o n s and c r o s s c u t s o f the s e p a r a t e l a t t i c e s f o r Hn and H 2> o r f o r L± and L g . F o r example, i f c o r r e s p o n d s t o « F( «C , Pi") and H 2 c o r r e s p o n d s t o , % ) , and i f the f o l l o w i n g automorphism elements a r e d e f i n e d ? under s, P/£ »-under t , »- ^ r ... r , a p r i m i t i v e r o o t mod p j under u, Sb" *~ @3fc, under v, ^ *"^ w ... w, a p r i m i t i v e r o o t mod q j t h e n the f o l l o w i n g c o r r e s p o n d e n c e s a r e a p p a r e n t : 1 x i ; i F(oi. ,@, %), {s} x l J 3%), 1 x{u} C I P U ^ ^ ) , {s> x{u} t ? F { o t , ^ ) = F ^ ) , {s,t} x 1 H ; F{/2, $>) = L g , 1 x{u,v} T F ( < * , 8a") = {s,t} x{u,v/ t ; F , e t c . t 1 e t c . The G a l o i s group o f o v e r F c o r r e s p o n d s t o G/H-^  The G a l o i s group o f L 2 o v e r F c o r r e s p o n d s t o G/Hg By g e n e r a l i z i n g t h e d i s c u s s i o n o f theorems 6.1 and 6.2 i t can be seen t h a t t h e f i e l d F(©c, ,o( ,o(_, .. .o< ^^^^1 P n / 1 2 3 n x c *.. v**n) * n a s over F a G a l o i s group G 7 x G o x ... x G, p i / — where G ± i s t h a t o f F£<. „/a ) , where ©Cr i s a p r i m i t i v e it I * P ± t h r o o t o f u n i t y , and where p , p g ... a r e d i s t i n c t primes. n1 -20-CHAPTER SEVEN The Galois Group of the Field F(^3, % +at-¥a) Consider the equation (x q - b ) p = a where p and q are distince primes, and a and b are rational indeterminates. The discussion to follow w i l l p artially collapse i f a i s a perfect pth power of a rational and w i l l further collapse i f b + l a i s a qth power of a number in the f i e l d F(B£). The same remark must be made for the conjugates. Hence we consider the general case where a and b are rational Indeterminates. Let o(be a primitive pth root of unity and @ be a primitive qth root of unity. Adjoin to the rational f i e l d F a l l the conjugate roots of the given equation to form the normal f i e l d F ( ^ * % + Jt'^fe), for i - 1, 2, 3, ... p and j - 1, 2, 3, ... q. This f i e l d K certainly contains c<, @ , and 3b + but by merely adjoining these four elements to the rational f i e l d F we do not necessarily get the whole f i e l d K, for we have no guarantee of the presence of the radicals % +o<W, 3b" etc., because extraction of roots i s not one of the operations defined in.a f i e l d . But by adjoining a l l radicals % + o^ 1*^ for a l l I • 1, 2, 3, ... p, as well a s ^ , we do get the whole f i e l d K. We wish to determine the structure of K and that of i t s Galois group. Theorem 7.1 The Galois group of K over F has order -21-pq p(p-l)-(q-l) and i s generated by the elements s, t, v, u^3 u 2 ... Up subject to the (sole) defining relations: s p B t p - 1 B u ± q B v q - 1 B l , where represents any of ^ 1 ' ^2 * * * ^ p' st B t s r , where r is a primitive root modulo p«, u 4 v B vu^ w where w i s a primitive root modulo qj - i i u i + 1 B s u^ s and s u ^ + 1 B \X^S^ SV B VS 5 ts = vts u i u j . • u j V u 1 + 1 t B t u r l + 1 . , where the subscripts are reduced mod p. (For each there exists the relation'u^v = V U ^ W where w i s any primitive root mod q. It makes no difference i f various primitives are used for w in the separate u^. The generators above are not an independent set since each u i + ^ i s expressible in terms of u^ and s. The four elements s, t, u^, v form an independent set, but the discussion i s developed more easily i f the additional u^ are used. Proof: Consider the automorphism of K defined as follows: Under s, PiT »-under t, <*. *- c*,r, with r a primitive root modp; - under U ] L, 3b + 9a~ + 9a"; under U«J under u ± + 1 , 9b + c C 1 " ^ »- £'9b + o^ -P/a"; -22-and under v ^ *- w> a primitive root mod q. As in the previous chapters i t i s easily shown that SP „ tP-l•„ ^ q = v q - l = i9 t h a t st = t s r and that U j V • v u ^ . But the other relations require investigation. Under s l U j S * - , while . o -^'P/a" »- P/S *- e*f-P/a*. But gives the same effect. Hence s" 1u 1s = u i + i * B y a s i m i l a r procedure i t is easily shown that s u i + 1 = u ^ . Now s and v are completely dissociated automorphisms; for under s, while/^-% + o k 1 * /3J-$> + , * 1 + 1 . P £ for a l l j , i j whereas under v,fl » ^ w . Hence sv = vs. Similarly, t and v are completely unrelated operations; since under t,o<—*-c<r} whereas under v,^ Hence tv •= vt. Again,since and u^ operate on distinct elements they are dissociated; for under u^, $b + o C 1 - ^ + o C 1 - ^ while under u^,$> + oC1-^" -+ erf^-P/Si. Therefore ^ U j = U < J U I * Finally, under both u 1 + 1 t and t u r l + 1 , % + oC1- P/a~ + ot 1* P/£f while ot 1 - o t r l and for a l l n # 1 , ^ - % + c * n - S & ~ — + 0 c- n r-P£ for a l l j . The automorphisms already l i s t e d (there may be more)^ with the stated defining relations generate a group of order pq p«(p-lMq-l). Form a l l possible products of the form s c t d u 1 e ' \x^z ... u p p v f . The order of the group w i l l then be the product of the orders of the separate elements s, t , u^, u 2 , ... u p , v unless there e x i s t more ( r e s t r i c t i v e ) r e l a t i o n s . No new elements are formed by m u l t i p l y i n g on l e f t or r i g h t by s, t , ^  or v, a l l the elements of the form s ^ ^ u ^ ' u 2 e * ... u p eP-v f, because commutation r u l e s have been developed f o r any p a i r of elements. I f any f u r t h e r r e s t r i c t i v e r e l a t i o n e x i s t s i t can be w r i t t e n i n the form s c t d u , e ' u 0 e * ... u e p v f 1 . Now 1 2 p si n c e s maps ^a onto o$£, whereas t , u^, Ug, ... u p » v a 1 1 leave Ba i n v a r i a n t , we must have s c = 1, whence c==0,mod p. Then t ^ ^ 1 u 2 e z ... Up^v^ = 1. But sin c e v and a l l u ± leave _r d oC i n v a r i a n t while t maps c<..onto ©c. , we must have t = 1 and u 2 6 ' u 2 6 , z •** u p 6 p v f = 1 # Continuing i n t h i s manner i t i s seen t h a t no f u r t h e r n o n - t r i v i a l r e l a t i o n of the form s c t d u 1 e ' u 2 e * ... u p e p v f = 1 e x i s t s . P r o v i d i n g no other automorphisms of K e x i s t , these generators s, t , u^, Ug,...up,v form a b a s i s f o r the G a l o i s group i n the sense t h a t every element can be w r i t t e n i n the form s ^ ^ u ^ 1 u ^ 4 ... u p e p v f ; but only the elements s, t , u^ and v are Independent. We now have the order of the G a l o i s group » pq p. ( p - l ) - ( q - l ) . There remains the task of showing the G a l o i s group can have order no l a r g e r than pq P ' ( p - l ) - ( q - l ) . To do t h i s , examine the whole f i e l d K constructed i n the f o l l o w i n g manner. A d j o i n ©c t o F g i v i n g F(«c) of degree ( p - l ) over F. A d j o i n /& to F(s*-) g i v i n g F{°c,j8). Since j3 s a t i s f i e s an equation of degree (q-1) w i t h c o e f f i c i e n t s i n F, then F(<<,^ ) c e r t a i n l y cannot be of any higher degree, over F(oc),than (q-1). Then F(<*,y# ) has degree ^ ( p - l ) - ( q - l ) over F. -24-Adjoin Sa~ to P ( o t , ^ ) . Since 5a~ satisfies an equation of degree p with coefficients in P, then Ffrj j^P/a") has,over P,degree ^ (p-l)*(q-l)(p). Adjoin % + Wa, 3b +o$a", etc.. Each satisfies an equation of degree, q over F(<*,^ 3, hence the degree of the whole f ield4pq p(p-l)-(q-l). The whole f i e l d K can he written either F ( c < , for i m l , 2, 3 ... p, or + of:95) for i = 1, 2, 3 ... p and j = 1, 2, 3 ... q; because each contains the other. This limitation on the degree of the f i e l d combined with the fact that the Galois group has order ^ PQP,(p-lM.q-l) establishes both degree and order as - pq^(p-lMq-l)* and the group i s well defined. The correspondence of groups and fields is quite complicated because the lattice contains several of those of chapter five as sub-lattices. A complete correspondence analysis could be made by taking a l l possible subgroups generated by s, t, u , v with generators taken one at a time, two at a time, etc., and then determining the corresponding subflelds.. The lattice diagram on the following page is only a skeleton lattice of the major fields and groups, and their conjugates. Since this diagram i s not complete the network does not imply that unions and cross-cuts are given, but merely the "contains relations" and grouping of major conjugates. PARTIAL L A T T I C E O F T H E F I E L D F ( ^ , y b + ae-.$6)--Kv- P-i A N D I T S G A L O I S G R O U P O V E R \tx a l l i > t * u, t,v, a l l u L t , u v u , a l l u L * u , t, a l l U ; * u, F ( £ a l l U j ^ u , F M y^m) fcuvu, a l l U i V u , t , v, a l l u ^ u 2 F » £ v , a l l u L st,u,.u z .-u P >v P ( r a r i o n a l s ) s"t5, v , a l l u L t ^ U j V u j . a l l u ^ U i etc-t , v, a l l u L # up F ( ^ P 5 e ) t , a l l u ^ u z Fft 36^551 - - - - e t c . - . - -a l l u L ? u z Fft ^ T ^ ^ K s t s ; ' y a l l u { F K ^ ) R.UpVup.allUii'Lif t , al l U;# Up a l l (J;* u, KEY G R O U P G C N E R A T O R S C O R R E S P O N D I N G S U B F I E U D /Co fLo&Lc'Ur' BIBLIOGRAPHY Artin, Emil - Galois Theory, Notre Dame, Indiana, 1946. Birkhoff, Garrett and MacLane, Saunders -A Survey of Modern Algebra, Harvard University, 1947. The MacMillan Company. Carmichael, Robert D. - Groups of Finite Order, University of I l l i n o i s , 1937. Ginn and Company. Dehn, Edgar - Algebraic Equations, Columbia University, 1930. Macduffee, Cyrus C. - An Introduction to Abstract Algebra, Wisconsin University, 1940. Wiley and Sons. 

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