DETERMINATION OP BASES FOR CERTAIN QUARTIC NUMBER FIELDS by Norman S a f f r e y F r e e A T h e s i s s u b m i t t e d f o r the degree o f M a s t e r o f A r t s i n the Department o f Mat h e m a t i c s The U n i v e r s i t y 0 f B r i t i s h C o l u m b i a A p r i l , 1939. D e t e r m i n a t i o n o f Bases f o r C e r t a i n Q u a r t i e Number F i e l d s I n t r o d u c t i o n . T h i s t h e s i s d e a l s w i t h the d e t e r m i n a t i o n o f "bases f o r i n t e g e r s o f the q u a r t i o number f i e l d d e t e r m i n e d by- the e q u a t i o n w h i c h i s i n the normal form, and where Qi ^ and vS" a r e r a t i o n a l i n t e g e r s . The method u s e d i s t h a t d e v e l o p e d b y If. R. W i l s o n i n h i s paper " I n t e g e r s and Bases o f a Number F i e l d " , p u b l i s h e d i n the T r a n s a c t i o n s o f t h e A m e r i c a n M a t h e m a t i c a l S o c i e t y , volume ,29* 1927 e F o r an i n t r o d u c t i o n to the problem, an e x p l a n a t i o n o f t h e n o t a t i o n employed and a s y n o p s i s o f W i l s o n ' s r e s u l t s , the r e a d e r i s r e f e r r e d to D. C. Murdoch's t h e s i s w hich may be f o u n d i n the U n i v e r s i t y o f B r i t i s h C olumbia L i b r a r y . I t would be w e l l to m e n t i o n h e r e one o r two i d e a s t h a t w i l l be u s e d t h r o u g h o u t t h i s e n t i r e work. F i r s t : The n o t a t i o n i s u s e d to e x p r e s s t h e f a c t t h a t ex- d i v i d e s ^6- , and a. £> f o r ci- d o e ^ n o t , d i v i d e . ^ . Second: The e q u a t i o n ^^-J-Q yr*-t-/?x -t- 5 a s a i s assumed to ^be i n t h e n o r m a l f o r m , t h a t is,<$, and %S a r e a l l r a t i o n a l i n t e g e r s , and t h e r e e x i s t s no prime /> s u c h t h a t y O " " V ' where •/» * , and f o r each o f /, s/ 3„ I n terms o f our e q u a t i o n , t h e r e e x i s t s no prime Z3 s u c h t h a t ^6 */*,<°S/S? and /*YS. The Bllm'inant and. D i s c r i m i n a n t o f the F i e l d , The e l i m i n a n t o f the e q u a t i o n i s g i v e n . h y Murdoch, page 6 , and i s 3 v- s' ^^-^5^^-^ 3 ^^o^,^^ -/ d/H'J^o < J < = £ ' - d ? * S ^, % ^ ^ i s an i n t e g e r o f the f i e l d ' - y * * + S = <^ t h e n the f o l l o w i n g .are a l l r a t i o n a l i n t e g e r s . where and hence - 6 - • F u r t h e r m o r e , i f J~, , , J-3 a n d ^ a r e r a t i o n a l i n t e g e r s , t h e n j f i s an i n t e g e r o f the f i e l d . ' The d i s c r i m i n a n t o f the . f i e l d e q u a t i o n i s src.) - 7 ~ Then from the d e v e l o p e d t h e o r y , i f ^ 6 " i s the h i g h e s t power o f o c c u r r i n g i n the denominator o f any i n t e g e r , t h e n The'work o f d e t e r m i n i n g the maximal r e d u c e d i n t e g e r o f the f i e l d i s d i v i d e d i n t o f i v e c a s e s as f o l l o w s : Case I: To d e t e r m i n e the maximal r e d u c e d i n t e g e r s o f the 1 s t degree i n #. Case I I : To d e t e r m i n e t h e maximal r e d u c e d i n t e g e r s o f t h e End degree i n X when none o f the 1 s t degree e x i s t . Case I I I : To d e t e r m i n e the maximal r e d u c e d i n t e g e r s o f t h e 3 r d degree I n X when none o f l o w e r degree ; e x i s t , Case IV: To de t e r m i n e the maximal r e d u c e d i n t e g e r s o f the 3 r d degree i n X when t h o s e o f the Snd degree e x i s t h u t t h o s e o f the f i r s t degree do n o t . Case Y: To de t e r m i n e t h e maximal r e d u c e d i n t e g e r s o f the 2nd and 3 r d degree i n X when one o f the 1 s t degree e x i s t s . The wo:rk o f t h i s t h e s i s d e a l s w i t h c a s e s I and I I . Case I . I f t h e r e i s a maximal r e d u c e d i n t e g e r o f the f i r s t degree i n ^ i t i s o f the form r _•/"'/*. w h i c h i s i m p o s s i b l e s i n c e the f i e l d e q u a t i o n i s i n the normal form, i . e . 'f>^ >? ~<9 f o r each, o f Si 3, -V • Moreover, -2~F ^ tr^ j o r i f £' i>^' 0 t h e n /> =(#o t ^ 2^c0j l e a d s to a s i n g l e - p r i m e r e d u c e d i n t e g e r f o r w h i c h /b^'/gdt • whence fi—^j ^ % • H e n o e ^ r e d u c e s to ( ^ » ) where 2. • "(For the e n u n c i a t i o n s o f the v a r i o u s theorems quoted, the r e a d e r i s r e f e r r e d , t o Murdoch's t h e s i s , pages 1-6, and f r o m —Z. s £4 < -L ~ — ^ / -7 T e s t i n g t h e n -10-We c o n s i d e r the c a s e , f i r s t , a0 = / • On s u b s t i t u t i o n i n JTy we g e t (i - i n t e g e r ) 4 a JT^ s a t i s f i e d i d e n t i c a l l y . C o n s i d e r i n g now the c a s e =»""/, we g e t , on s u b s t i t u t i o n , . -4/ - / f - O J s u c h t h a t S i n c e t h e r e e x i s t s a maximal r e d u c e d i n t e g e r i n ^ o f degree 2., i n 7L , t h e r e e x i s t s a s i n g l e prime r e d u c e d i n t e g e r o f t h e f o r m = ("or /J • S i n c e y i s an i n t e g e r so a l s o i s y 7t R e d u c i n g bjr means o f f i e l d e q u a t i o n f x * = ^ «i ;*°V ^- ^ V ^ ^ s/ /* jh — ^ a l s o s=> /=> /*> / b y o -6 — - x — i s an i n t e g e r A l s o - 15 -Hence the congruences We n o w - d i v i d e Case I I i n t o 8 sub-cases, 1. / V 61, />//?, J5 . . 3. yfy J , /*/.•$ , J X ,/> . A. 5 . 7. - 16 -Case I I . . Sub-case 1 f>//?, /Oy6$. I n t h i s c a s e i f t h e r e i s a maximal r e d u c e d i n t e g e r o f the End degree i n X t h e n t h e r e i s a s i n g l e prime r e d u c e d i n t e g e r , (aor /J and su c h t h a t theorem A r e d u c e s t o 2 S i n c e f> Jf>j)-T e s t i n g ^ ~ £ a n a f>i= 3 we g e t a<3 = ^ h^>-<*< which i s i m p o s s i b l e . Hence ^ =- 3. • \ T e s t i n g t h e n ( , o, f j ? * - / , C L , =o s a t i s f i e d i d e n t i c a l l y . Hence t h e r e i s an i n t e g e r o f the form jf) • under t h e f o l l o w i n g c o n d i t i o n s I f i s n o t maximal we t e s t f j ^ ? / / v ^ j where ^ - ± / . T a k i n g the case <* C o n s i d e r i n g (1) «? ^ 3 / - ° / 6 (3) v? S o ^ 7 ^ > ^ f / f . 20 -From (3)Y(n (3J These c o n d i t i o n s s a t i s f y J T s , -^J and J ~ . These c o n d i t i o n s a r e i m p o s s i b l e , s i n c e by h y p o t h e s i s , t h e r e i s no maximal r e d u c e d i n t e g e r o f the f i r s t degree i n ?( . T e s t i n g now <*0 - ~(i ( ' i > ) • w i t h -S = J' -h /? - ? s btit - 21 -Now from ^ -+ / f ^ =.a Hence, now i s I d e n t i c a l l y s a t i s f i e d by ^ 4 . + $2+2S «*~S °' 2*) . S i n c e the e x i s t e n c e of. i m p l i e s t h a t o f ( ^ , we have We c o n s i d e r f i r s t JLM ":. i d e n t i c a l l y s a t i s f i e d , l e t •J r e d u c e s to i . e . J? (_/ +.q +sj a~-/? Q ~o ^^*s#s~£ r h i c h w i t h the above, c o n d i t i o n s , t a k e n f ^ f ^ f /6 o n t r a d i c t i o n . Hence t h e r e i s no i n t e g e r f o r C o n s i d e r now = -t~/, ZZ/: i d e n t i c a l l y s a t i s f i e d . — " -* . )• V ?~h'-*a 6^ w h i c h g i v e s V f . which w i t h . /? « ^ " / f j , . Q =. i- At & s g i v e s L e t /Pj -Then _ Z T g i v e s = <2 # • ... and hence /? 3 = — - ? ^ o - ^ %• Hence jZ^ a r e now s a t i s f i e d i d e n t i c a l l y and we have the i n t e g e r O, - ^ J the c o n d i t i o n s f o r i t s e x i s t e n c e LF- C ^ r 0 i i s n o t maximal, t e s t t h e n J- M L e t . ;? ^ ~ J^ray s a t i s f i e d i d e n t i c a l l y . ^ 6 w i t h A « < a ^ ^ - g i v e s amongst o t h e r c o n d i t i o n s t h a t 6) s o which c o n t r a d i c t s our e a r l i e r c o n d i t i o n s . Hence'/^sr/ °'j-»J i s maximal. We c o n s i d e r now the g e n e r a l i n t e g e r , / " ' ^ Q where 3, 2 *t W r i t e •» , „ -J3 ~" J?**~ We s h a l l d i s c u s s JT^ f o r -»*=J and s i n c e t-- af^^*~*~<*>/o w e g e t On s u b s t i t u t i o n we g e t from jr- ; i d e n t i c a l l y - s a t i s f i e d . 3 • Since, s /'4 and M**'^ , we r e d u c e to F u r t h e r , l e t .j #, f s J S r £ J ^ a * 3 ?hen * - S? - * 4 S $ - 4 /? a . 3 ^ « s - ^ s 3^ t a k e n modulo ^ g i v e s L e t = -*•- ^ + 3-^\ i n a g a i n we : g e t ^ £ = /?, = s, -h 4 , ? . i .e, J., r e d u c e s to S,<*,.+Q , Q i s s a t i s f i e d by • f 0/ y J i s an i n t e g e r when /? = / -4 3 O S •£ O 3, a 6 i t + / - 29 -where * *" ' * 3 a •= 3 4 , , Next we c o n s i d e r = - / Hence we have ^ ^ ' / ^ V 3 and j v ) i s maximal when ( 01 ~g , —-jr J i s maximal when 61 = - 3 A ^7 /? = x • + T A Case I I . . Sub-case 3 » I f t h e r e e x i s t s a maximal r e d u c e d i n t e g e r i n yo of the 2nd degree i n X t h e n t h e r e i s a s i n g l e - p r i m e r e d u c e d i n t e g e r y ^•••^(•a0, /J s n e h t h a t theorem A becomes' ( 1 ) (2) -t- 4 ) = » ^ y i From STj-faj we g e t on s u b s t i t u t i o n < -t- 3 /? =• ^' \ '. e i t h e r — ;? or ^ =' <2 s i n c e yb ^ • and hence i f y£> ^ 2 we - have. -t- 3/? =0 f^-W^c s i n c e S i n c e J^tf we have <$ s. ? and s i n c e ?/ / f y reduces to <2 / V - / ? , 2* ..which w i t h 6i - / reduces to and w i t h _ Z j reduces t< - 37 -w i t h 5 = ^ -<* * we would r e q u i r e / 4 which i s c o n t r a r y to our hy Hence (a, , ~L ^ ±s maximal when - 3 9 -Case I I . Sub-case 4, From Th. A, i f there i s a maximal reduced i n t e g e r i n /> of the 2nd degree i n ^ , then there i s a s i n g l e -prime reduced i n t e g e r , J~ jg-(o( Ot/-Cf0 +61) 30 ^.>-x^-W Now O i , = £ 0 s i n c e i f CL(-O from (2) we would get Jo //? which i s I m p o s s i b l e . From (1) e i t h e r //? . Therefore c ^ ^ O i m p l i e s rt/ j£ O• t ^ T ^ f l y ^ } t h e n f r o m (2) /fa-^a,. ( i i i ) I f ac r- et,a-*o *o ^~o-<*/o • then from (2) ^ S a ^ ^ ^ , which i s i m p o s s i b l e . Then c o n s i d e r i n g the f o l l o w i n g cases ( i ) ctQ •=>aj y a, ? 0 . From Test then ( o ( . Then from The. A. above, / f 2 # ^ - 40 -From , , -^3T« j 2 * i m p o s s i b l e s i n c e Hence t h e r e i s no i n t e g e r o f the is —. / (t x. / i i ' n "this s u b - c a s e . J - ( ( cP ' * J • ( i i ) hyp. ^ Then from 2 Th. A. T e s t i n g ~ f > ^ -g i v e s , i f <3 <*o s A and from t h e n ft - ~ ^ From T. we g e t t h e n w h i c h i s i m p o s s i b l e s i n c e each <=< was supp l o w e s t terms. - 41 -* i Hence we do n o t have an i n t e g e r o f the form T e s t yo ^ iL. -T = O r > i r < 5 ) . From Th. A, t h e n 6( -^a w h i c h i s i m p o s s i b l e s i n c e 2 . Hence we can have, no i n t e g e r o f the form ) Hence t h e r e a r e no maximal r e d u c e d i n t e g e r s i n o f the 2nd degree i n x i n t h i s s ub-case y?J<%,yo-J/?f / J . - 42 -Case I I . Sub—case 5. I f there e x i s t s a maximal reduced i n t e g e r i n /° o f the 2nd degree i n % , then there e x i s t s a s i n g l e prime reduced i n t e g e r u= A , _ /1 such t h a t Theorem A reduces to ( 1 ) etc ( ^-^o 5 ^ We cannot have cf^ — o f o r i f c?a — o we get from (1) .S = O c ^ o - v ^ - o ^ y/o which i s i m p o s s i b l e . We then c o n s i d e r the two cases. (a) d 0 O, a , =0-(b) ci0 ^*c>, 3 - 4a<> 4 -+ A{Q*-(-fiS)ao ^ t y f 4s which reduces to and:since i . e . We f i n d i s i d e n t i c a l l y s a t i s f i e d hy Hence we have the i n t e g e r ^ / under - 44 -under the f o l l o w i n g c o n d i t i o n s and where a0 i s so chosen so as to s a t i s f y m aximal } t e s t Now i f / QJ. o i ) i s n o t .'• f ^ <3x ~L v S i n c e the e x i s t e n c e o f t h i s l a t t e r f o r m i m p l i e s the f o r m e r , we have the o r i g i n a l c o n d i t i o n s s t i l l h o l d i n g , i n p a r t i c u l a r T e s t i n g / Go / a.r_ _z_ i we g e t from and s i n c e Now s i n c e /? = 0 ^t^ofyt, 3 , and =f ? » w e g e t * <3( - ^ d< * • • ' ' ? C?/ Gases (1) ^ . C o n s i d e r i n g Case ( £). Q z :s 2. 5 From 6( z •= M $ i^o-of we get 5 =.o ~^*}~*#-o( / o j which i s contarjr to our h y p o t h e s i s . T h e r e f o r e , we cannot have a( • C o n s i d e r i n g Case ( 1 ) , =* & • Consider then / a a 0 . ~L \ . On s u b s t i t u t i o n i n and s i n c e ^ ^ ^ ^ ^ = cV which w i t h JT~ reduces to which w i t h g i v e s - 46 -/V if Ct /of -32 then f Now, s i n c e ^ ^ 3" i . e , Hence ^ / i f o l l o w i n g : c o n d i t i o n s i s an i n t e g e r under the and where a0 i s chosen to s a t i s f y If / &o * J_ \ i s n o t maximal t e s t - 47 -S u b s t i t u t i n g i n 2 #6 3 ^ but ^ £ ^ Hence we t e s t / . / \ We t e s t now the g e n e r a l i n t e g e r - > 7 - £ ~ $L^ " 4 O., 1-3/^ = <3 ~^y^C*-~e{/k> Now t h i s c o n d i t i o n w i l l be t r u e f o r ~ 3 *f (t O . T h e c a s e /-=0 j S±res O L C = 0 . C o n s i d e r f i r s t ^ = .jf f-^ $ . Hence from w e g e t and s i n c e /o~# 6( J ^ A - 48 >-l l s o ^ =-3, £~ ==• Z g i v e s hence _ s i n c e £ Q We t h e r e f o r e c o n s i d e r the g e n e r a l i n t e g e r 3 «0 —<^~?~^<34f>'^1 r 3 : _£ - » s i n c e 2 - 5 x > , -r- , j Z - ~ „ * -> Therefore ( * ± ^ i 8 m a x i m a l w h e n and where ^ i s chosen to s a t i s f y Case (h) ^ , , T e s t i n g J/ ^ ^rC^Of ' 'J we get from S i n c e y> -£ 2 and y , — O f which c o n t r a d i c t s our h y p o t h e s i s . Hence th e r e i s no i n t e g e r i n t h i s case. , Consi d e r now the case when y, We can have the f o l l o w i n g p o s s i b i l i t i e s : - 5G if'0' w M c h i s ruled-out' by (1) of Theorem A which g i v e s Q * / - 3-3 ->^—^— a which i s i m p o s s i b l e s i n c e a., C j ) = /? f We c a n n o t have ~ o f o r from (2) we would have ^ -a.a ^ ^ ^ y ^ w h i c h i s c o n t r a r y t o hypothesis,, A l s o we cannot have aa ~ o s i n c e from (1) 3 o i*~*~oc, a l s o c o n t r a r y to h y p o t h e s i s . The o n l y case to c o n s i d e r i s a ^ ^ =.^Jfov j/ -.^ 0{ aj. / uL-j g i v e s from 7"* 2 4 == 6i m.0 Z 6 " /o * We have as c o n d i t i o n s on . ( e , ^ ^ ^ r y S . S u b s t i t u t i n g t h e s e c o n d i t i o n s i n / , g i v e s ' A * /**.5 T ' - ^ X S z ? 5 y- 5 a These c o n d i t however, are i m p o s s i b l e s i n c e the equa t i o n form. i o n s , was i n the normal Hence if/o.?-X the o n l y p o s s i b l e i n t e g e r s of t h i s form are: jr. -Cp , a t , y j , a, +e, „, T e s t i n g s'Yx, • have as c o n d i t i o n s from p r e v i o u s t e s t t h a t S u b s t i t u t i n g f i n L e t ^ J / " * • * * " which, w i t h reduces to — — V s *, yy 2 / S reduces tc and f u r t h e r w i t h jo/6L.-,f7A, r ' 4 -t-KZ-qqS + s^a0a.( -4l + /6 o.( V / f « ( i y / ? 3 o -t^>~^ p - 58 -U s i n g /=/&., /° *//?., f •*/.•$ . . . and t a k i n g modulo /o6 g i v e s C a s e s : (1) — O i m p o s s i b l e by h y p o t h e s i s s i n c e - 5 9 -which from _Z~ : ^ „ . ^ . g i v e s ct0 — o c o n t r a r y to h y p o t h e s i s . s ^ c ^ X +6o.,r*fti Hence from ; 3 _ ^ ^ ^ o -g i v e s cra p c o n t r a r y .to h y p o t h e s i s . t h e r e i s no i n t e g e r of the form which w i t h J\ " < p i > T e s t i n g ss.t Q ( ^ ^ ^ ^ ^ ^ ^ as a l r e a d y e x i s t i n g c o n d i t i o n s ^ / $ ) / t o * / /?. S u b s t i t u t i n g ^ - ^ i n _2- too - iA. = c i f where ^ <$, .'.-.} We cannot have y ^ . s i n c e ^ f^C. ~~/«"< / 3 ff/^ 4 d/'-i^^S ^ ^ s ^ * s a ^ ^ ~ * 2 t-J s, • — - — /*' \ ^ />< o which reduces to 3 - 61 -/~A~ ~~/»* /°? M ~~ 3 A <% S i~'6 $ =0 - ^ r u i u e f / o ° which w i l l reduce to • •'• f $t fi* -*^<-^/= * Cases (1) 6(( ~o ^yruf^of ^ j out s i n c e ^ o^^f 6(. (2) ~-a ^s^yj^c/yrz. i . e . / f =^ ^ry^o^/o * which w i l l s a t i s f y 1^ and Z^. Hence £ 12-, °' A>)' ^ S a n ^ n'' : e^ e r w k e n and i s chosen so t h a t Hence i f . ^ 2 the o n l y p o s s i b l e i n t eger i s ( .y, '. u< fa> ) r a0 -^o . I f t h i s i s not maximal , there i s an i n t e g e r of the form / a& Q. where ^< 3 . T e s t i n g n o w ^ = , a / _^ w i t h the a l r e a d y e x i s t i n g c o n d i t i o n s ^ ' S u b s t i t u t i n g i n which w i l l , w i t h i- reduce to 3—rr ,5> /»- « 7" • &o ~/f a, 1- ^^4% +s'4t /**,/>*.4,*/?* +U*,^"Ji 4(/?j J>*f>7/?s 3 a * , * ^ , ^ - 3 * ^ a ^ ^ s ^ _ ^ ^ / G r where from j - ~ we had 7 * - 65 -Talcing now g i v e s : w i t h J - , ^ - r where from our p r e v i o u s c o n d i t i o n s - f o r . / " ^ . ' ^ ^ , we had ^ * / / f . ^ ^ and s i n c e jL Cases (1) (2) tf, ^ i m p o s s i b l e s i n c e <^ C3) / f ^ * and .2^ - 66 -A g a i n m 2~ we get and s i n c e (1) 6> = tf (2) we have as cases •ay/to-Hence w i t h we get which g i v e s (1) or, = and sin c e - y> (2) Now ~2f-4, ~4 K ^ T h i s , w i t h „ . . g i v e s w h i c h w i t h (2) g i v e s w h i c h i s i m p o s s i b l e . Hence t h e o n l y p o s s i b l e i n t e g e r i s S u b s t i t u t i n g t h i s i n 1^ we g e t J. L e t -A 7— 3 3 *7 b u t 5 -o* - 6 8 now y # /? .v- j / ^ > ^ / f l e t >»f-Cases. (1) _ then / f ^ s ^ 6 (2) ^ from ^ ; ^^yy> requires.• ^ .s.^^^.y^ which i s i m p o s s i b l e . Hence / i s an i n t e g e r under the f o l l o w i n g c o n d i t i o n s : p — and .where ^ i s so chosen t h a t We now c o n s i d e r the case ^ = p I f 2L- i s not maximal then there I s an i n t e g e r of the form /ae _z I t was shown p r e v i o u s l y t h a t we cannot have both ^ ^ and ^ —O. There a r e , t h e r e f o r e , three p o s s i b i l i t i e s (a) &> • (b) a; , ). ^ ^ a / J^O-T e s t i n g fe, ^ , w i t h the a l r e a d y e x i s t i n g c o n d i t i o n s 3 / 4 , ? /?( 2 V -5 • m c e i d e n t i c a l l y s a t i s f i e d . £ <*/ —/A 4 -&4/4 -/-.a * y-^y? =^ 7" - 7 ^ ^ ^ ^ = ^ "Again i n 2-# 4, /- 4t -tzs. ^-O ^ J ^ ^ ~* - 0 hen f r o m -r-^ ?*" c?" •? 7 *?* j?r z>*~ ^< /4 t4/s4; -J* +t**3 +/a r/?3 ^ A g a i n i n x~, g i v e s 3-T h e n ^ ^ ^ j i s an i n t e g e r under the f o i l o w i c o n d i t i o n s : ng where 4 6>. of the form lf(£'~j'^>) i s no't..-maximal there i s an i n t e g e r where S u b s t i t u t i n g t h i s i n 7 - • ' *7/ - 7 4 -r4 4 3 3 4 *<• s w i t h <07 I n p a r t i c u l a r taken T g i v e s 2 — * ~^***-//?, /o^vS C o n d i t i o n s E ^ S u b c a s e 2. J--x + y,S •3 J 77 —r? ' >ubcase 4. <3, /° 2 ft / V v 5 Subcase 5 „ There can be no maximal r e d u c e d i n t e g e r o f the 2nd degree i n t h i s c a s e . -^ -z i s the g r e a t e s t i n t e g e r f o r which t h e s e congruences h o l d and where a0 i s chosen so as to s a t i s f y aa0 = & -<^^/=> S S / /f, y ^ 5 » / I n t e g e r s C o n d i t i o n s Subcase 6 . There can be no maximal reduced i n t e g e r of the 2nd degree i n t h i s case. Subcase 7. / 3 ; and where a 0 i s chosen to s a t i s f y g ~ $ ^ ^ / b ^ si = & = s , ^ where C e x i s t s , but none of lower degree i n , then th e r e e x i s t s a s i n g l e - p r i m e reduced i n t e g e r such t h a t , - • (1) a a = ~S (3). a 7 - a , + ( by the f i e l d e q u a t i o n , we have S u b t r a c t i n g t h i s from a.a y we get V S /o S i n c e t h i s i s an i n t e g e r of the 2nd degree i n * i t must reduce to o r d i n a r y i n t e g e r s , and hence ( 1 ) -the theorem s t a t e s