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Determination of bases for certain quartic number fields Free, Norman Saffrey 1939

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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,<$, and %S a r e a l l r a t i o n a l i n t e g e r s , and t h e r e e x i s t s no prime /> s u c h t h a t y O " " V ' where •/» * , and f o r each o f /, s/ 3„ I n terms o f our e q u a t i o n , t h e r e e x i s t s no prime Z3 s u c h t h a t ^6 */*,<°S/S? and /*YS. The Bllm'inant and. D i s c r i m i n a n t o f the F i e l d , The e l i m i n a n t o f the e q u a t i o n i s g i v e n . h y Murdoch, page 6 , and i s 3 v- s' ^^-^5^^-^ 3 ^^o^,^^ -/ d/H'J^o < J < = £ ' - d ? * S ^,<x, «/+(4<k$ ^3 *l*J "Scot's ' We, h a r e , t h e r e f o r e , -_ A _ 0 A 3 ^ 0 *3 I n a l l t h i s work, and i t s s u c c e s s i v e p a r t i a l d e r i v a t i v e s w i l l he r e f e r r e d to as JZ, , -^3/ -3/ / -^raj ' ' -Esrcj » r e s p e c t i v e l y . To f i n 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 jr-^-f.6( xV/? Jf'-h.S = 0 we use the f a o t t h a t the d i s c r i m i n a n t o f any q u a r t i o i s e q u a l t o t h a t o f i t s r e s o l v e n t c u b i c . I n t h i s ease the r e s o l v e n t c u b i c i s J f J — * ^ - / ? * - o and hence the d i s c r i m i n a n t ^ i s g i v e n by - 5 -Fundamental R e l a t i o n s . I f ^ + ^ ^ - f ^ > % ^ ^ 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 _•/<?« J- i We. cannot have a0 - o , f o r i f so Jf ~^jt, ^  i SLn^ o n s u b s t i t u t i o n i n J2Tt , .j\ />"'/*. 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 <?9 ~o w h i c h was shown to he i m p o s s i b l e . By Theorem I I ( o ) , < £ / ^ £"Q and hence we must have 2T, — £~0 From Theorem IV ( c ) , s i n c e &o ^  O we have/*> /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 - <n - 5 a / - /e ft A-4 I f t h e s e c o n d i t i o n s a r e s a t i s f i e d we h a r e the maximal r e d u c e d i n t e g e r s — / - 11 -T e s t i n g now, On s u b s t i t u t i o n - i n 1 • © • j •3 . J.* : JL 1<L 9/1 as 4 /? 4 1* s 2 2 4 4_ » * 6l = - 3 ^/ F • Now -^j and we g e t ' • s a t i s f i e d i d e n t i c a l l y , . Hence i f the above c o n d i t i o n s a r e s a t i s f i e d we have t h e maximal r e d u c e d i n t e g e r ^ 3*""* T e s t i n g now- Z~- ' and from / y r e d u c e s to ( |2 , ^ . hence we t e s t Jf. ~ ( 3 ' a / • On s u b s t i t u t i n g I n •JL. ^3 2 6} 2* 3 * — 6--7 » J? a 4 3 Now -Z^ and g i v e s a t i s f i e d i d e n t i c a l l y . Hence i f the above c o n d i t i o n s a r e s a t i s f i e d we have the maximal: reduced i n t e g e r ; t l_ - 13 -T a b l e I,. .; Maximal r e d u c e d I n t e g e r s ' o f the f i r s t degree i n . ' J- H- X 0, = * ^r^-ct 2 To show;that t h e s e i n t e g e r s e x i s t , an example w i l l h e l p to i l l u s t r a t e . F o r example f o r (1) o f T a b l e I, we have the c o n d i t i o n s s a t i s f i e d by ^ •=/£) - 2 f • S * 3 33 . Q. - /O ^ ^ a o S". F o r ( 2 ) , we can have - 14 -Case I I . We f i r s t prove the f o l l o w i n g . Theorem A. I f a maximal r e d u c e d i n t e g e r i n o f degree two i n X e x i s t s , b u t none o f l o w e r degree i n X , t h e n 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 , J< ~ (A°' Q'> 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> J<S from _u #0 o . Cases T e s t J ~ /t<- /^ • S a t i s f i e d i d e n t i c a l l y s i n c e fi./&t / ^ • S i n c e ^ and i f -Z we g e t t h e n cfa s a *H*-etf* which i s i m p o s s i b l e . T e s t i n g now /o - 3 , i . e . , ¥ = (£A , ^ , \ . Now • <a0 ~ af — Theorem A, J- r e d u c e s to *C'-0 - ~ - ^ ^ - V ^ whi c h i s i m p o s s i b l e . Hence t h e r e can be no i n t e g e r o f t h e form (J->f>j)-T e s t i n g ^ ~ £ <BJZ , o, ^ g i v e s _ _ S i n c e y t > 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 <?b » / • y we get from, -ZT^ , i d e n t i c a l l y ' - s a t i s f i e d . J * Jt-f J1* From J"? f - z ^ ^ y above, w r i t e .5 ' -SA-JL +&-/' and t h e n r e d u c e s t o A l s o Qx-Z'.<kt, - ' -2Tj g i v e s ^ ) / = / ^ ^ u ^ 2, ^i-jz. and hence JHj r e d u c e s to 3 = w h i c h r e d u c e s t o - 19 -Now from /-t-^ -t-J^ s 0 2TS .* i s now i d e n t i c a l l y s a t i s f i e d . 1", •J1 ^ c ^ * c ? ^ 4L&. = * which w i t h r e d u c e s to Now rV-<f 4^ / f / ' j T^?^, s<s> <* 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 = <vk +Q -/ -•^H s a t i s f i e d i d e n t i c a l l y . ^3-jL. -/- i_£ 4*~t- a* J« f J?* / f- 2 4, V- 4, 7 -t it -t 2/?A =o r e d u c e s to , •+ ^ ~ o -n^o-** %> 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<U % ^ - V - " j h r + a I ^ < * * -Y-.S -+3. 6? /? =.0 r e d u c e s t o , W r i t i n g ^ £ = / - t 2 4 * - / ? J - j 4 * < (?3 sg. •  /p at^. a n 3 - s u b s t i t u t i n g i n g i v e s I f --^v i s "odd, ^ v=<?-^ ?.+ / we have even, and ^ even. That i s y3 / / Now J - 3 and a r e i d e n t i c a l l y s a t i s f i e d , w h i l e Z ~ , l e a d s t o a c o n t r a d i c t i o n . Next, c o n s i d e r even. Hence i s odd, and £R i s . odd. T h a t i s , /6 - 22 Now _Z~3. • and a r e i d e n t i c a l l y s a t i s f i e d w h i l e -27 r e q u i r e s t h a t 3 s <3 - / • I t i s more d e s i r a b l e to c o n s i d e r the work now f r o m the g e n e r a l i n t e g e r . T h a t i s , i f ( °' i s n o t maximal, t h e r e must he an i n t e g e r o f the form (' 1 Jr™) , £ < ^ However, b e f o r e we can d i s c u s s t h i s g e n e r a l i n t e g e r , i t i s n e c e s s a r y t o f i n d t h e c o n d i t i o n s f o r t h e e x i s t e n c e o f _y'~ (^' •> °' 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<n, t~3,/,o. F i r s t we c o n s i d e r •= • _ZT^ becomes ^ «•«, =* • f*-^\. X j becomes <$ frtf i"- * ' which w i t h = - / - i * - ' * - i 4 / g i v e s ^hv^ia/*f which c o n t r a d i c t s our e a r l i e r e x i s t e n c e c o n d i t i o n s . T h i s . i s m e r e l y our c o n d i t i o n t h a t be maximal.. C o n s i d e r i n g ; becomes and with. JG0 = <$ +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 w h ich 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 on •-. -Last, we c o n s i d e r - 5^ ^c^- a/ ~° j~3 "becomes now and JTH 2 = ^ T h i s r e q u i r e s amongst o t h e r t h i n g s t h a t w h i c h c o n t r a d i c t s e a r l i e r c o n d i t i o n s . T h i s i s m e r e l y our t e s t f o r the m a x i m a l i t y o f ( a ' ) - 3- t'~/-H e n c e 1 i n t h i s s u b - c a s e we have t h e f o l l o w i n g maximal i n t e g e r s . (1) ( fr*, f) . (2) (-f9, -f'fi). - 27 -Case I I . Sub-case 2. ^ /$, A 2 /°^ • 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 2nd degree i n X t h e n t h e r e e x i s t s a s i n g l e ^ p r ime r e d u c e d i n t e g e r y = -^{ae,atl /J s u c h t h a t Theorem A r e d u c e s t o Now f o r i f a., -a from 2, we would g e t /f s o fo^ei/o which i s c o n t r a r y to our h y p o t h e s i s . From (1) e i t h e r a0 =a, ^ * s - m ^ X / o b u t n o t b o t h . F o r i f % -o t h e n a, •= o which'we saw above was i m p o s s i b l e . The c a s e s to be c o n s i d e r e d a r e (a) o0 = <?, =t 0 , t h e n from (2)' /? S- /o (b) &0/ 0-f * O t t h e n a,* &<a0 ~^ru^*and f r o m (2) we g e t a. — aa &e S i n c e and s i n c e ^ w e . must have ^ ~ -3 Hence we t e s t y =fat , -£-J where a, - — /• We f i r s t t e s t .= /y ^ . ^. J-fa, Now f r o m ^>-- 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 </ - f , s a t i s f i e d i d e n t i c a l l y . w h ich w i t h /?• 3 / ? t — * r e d u c e s to i* — Q ( -T F JL 2 • r e d u c e s to /f / = — L e t /? = ^ -6(( + 3 ^ f and on s u b s t i t u t i o n a g a i n i n _/ ^ g i v e s From - 30 -and c s o A l s o from, /?f -a 5^ — <kt 1 « 6 » ^ 3^ s * 3^ $ 3 -h 5 y- •-S 3 s o L e t ^ =5' ?SA . t h e n we g e t . s ^ 3 o 1. e, •5 3 , Now _Z i s i d e n t i c a l l y / s a t i s f i e d w h i l e and hence In JT/ we ' r e q u i r e i . e . ^ s o ,(Of ~j~y J i s an i n t e g e r when - 31 -= (Si 3. S? 3 • — 00 at, T e s t i n g and s i n c e ^ /*» ^ T e s t i n g then the number we must have. J~ f( °ot / ) y where ^ = ^ = ± / From -J. we get on s u b s t i t u t i o n ± ±_ — si <k ^ 3 and s i n c e J /'<Q then ' ^ =<j 3 which i s i m p o s s i b l e . Hence th e r e can be no i n t e g e r ' o f the form y ^ j ~ C <?a, P , j /J . u n l e s s c*o — O • But from our hy p o t h e s i s /? s - fa °-t 3 w i t h = £ 5 : g i v e s y ? 5 a -^-w-*-**' 5 which i s c o n t r a r y to h y p o t h e s i s ybjl , The o n l y p o s s i b l e i n t e g e r s were .^<^  — , -4- ) . I f these are not maximal c o n s i d e r the form where J.. Y = / ^ -X 1 On s u b s t i t u t i n g i n - 32 Tfow s i n c e • i . e . 6( . 3 o J<ka< -f- £_/£ t a k i n g $ a, + A ft s <? and s i n c e f , / which i s c o n t r a r y to our h y p o t h e s i s . Hence there i s no i n t e g e r of the form / 2£. . -^r ) • f o r _ -z( f< Therefore j > 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 ) = <? We c o n s i d e r , f i r s t , the f o rm J- />» ^ 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. <?/ ^ ° " • We t h e n h a r e two c a s e s to c o n s i d e r • (a) ) b o t h f o r yo 3. (a) We cannot haye we g e t ^ 4* - <l a0 -=o and ^ =o f o r from But ^ 4 and ^ ^ ^ hence we have a c o n t r a d i c t i o n . From Th. A (1) we have t h e n - 3 5 -» Now from TZ. 4* go — 3. 6i Z 6 and s i n c e Hence, w i t h Q = a0 ^ ^ u / g i v e s Cfa -so -^y^o^ef yo , which i s i m p o s s i b l e . We t h e n c o n s i d e r the c a s e = ;? . W-e 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 : Lves We r u l e o u t ( i i f p'' # ) s i n c e (-1) o f Theorem A gi-i i . e . Q = o -^r^r-** H which i s i m p o s s i b l e . Uhe f o r m ( -£/°< ^ J i s ™^e& o u t s i n c e -/$ g i v e s 4 6 - 64 f ^ V j j =o w h i c h would g i v e w i t h o t h e r c o n d i t i o n s w h i c h i s i m p o s s i b l e . The f o r m • (• o, o, JL ) i s a l s o r u l e d o u t by j and s i n c e 2 / S / (Si ^ ^ o which i s c o n t r a r y to our h y p o t h e s i s . T e s t i n g now ( O / _Z~^ ; s a t i s f i e d i d e n t i c a l l y J.. i . e, $ 2 -t1- <$> -t- 3/? =0 f^-W^c s i n c e S i n c e J^tf we have <$ 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 = ^ -<*<S* Ta s o s a t i s f i e d i d e n t i c a l l y "by Q ^ Y * / ? H / ^ T h e r e f o r e / ' 01, J-^ i s an i n t e g e r when I f Z''7/ ^/ /* i s n o f j maximal t e s t then From 2 > * 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-(<?0, a , , /J such t h a t ( ] ) <>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 <x0 =o or cx, a - Q0 +-4 o ^i^a'. o r both. • I f ( i ) a a ~ o , then from ( 2) * + # J s / ? »~<^/o Therefore ctf-+0- i f a , = o we have yt>//? . 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>, <z{ (a) o 0 *&o( a , - a-We t e s t then the number / ^ / i On s u b s t i t u t i n g t h i s i n and now, i f i . e. - 43 -which w i t h , 33, J.. reduces to S > 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 <k. and <S are odd. (S) The form / g i v e s from Theorem A 2. s./ J.^: g i v e s : i d e n t i c a l l y s a t i s f i e d . 2* T h i s reduces to # ( s where J., which w i l l reduce to and w i t h which i s i d e n t i c a l l y s a t i s f i e d by 4, + V =< Hence t-fij i s an i n t e g e r under the above mentioned c o n d i t i o n s f o r M~(. If i s not maximal, we t e s t {'^T^y jr* J T h i s i n t e g e r i n J Q re-quires t h a t which i s c o n t r a r y to our h y p o t h e s i s y Hence the i n i i s maximal when teger / ^-. -C \ 6i =.r. - 52 -Case I I 0 Sub-case 6 , 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 s e c o n d degree i n r?C 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 -^(^ft a<s /) such t h a t Theorem A r e d u c e s t o ( l ) a ) ^2"> 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 ^ ^ <y • T e s t i n g j , - (J^L / From £ ~ = * i s an i n t e g e r o n l y i f ^ =*• . C o n s i d e r ^ f ) • From ^ - . a t f ^ i d e n t i c a l l y s a t i s f i e d , i m p o s s i b l e s i n c e ^ ^ ^ ^ ^ X 7-Hence t h e r e i s no maximal r e d u c e d i n t e g e r s i n t h i s c a s e . Case I I . Sub-case 7 , I f a maximal reduced i n t e g e r e x i s t s , then by .Theorem A, there 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 (2) a, {a, = aa a( From (1) e i t h e r , a. ) o r b o th. I n any case * g i v e s aa^a.( =. o . Hence, i f maximal reduced i n t e g e r e x i s t s , there e x i s t s a s i n g l e pi reduced i n t e g e r • On s u b s t i t u t i n g T g i v e s % & « 7 T and I f ^ =5* ^ ^ / ( % _ d< * # 4 3<? ' s s m o e I f . X* i s not maximal, there must'"be an i n t e g e r o f the form w e cannot hare both and a( = f o r (°( a< j g i v e s O ^ / O •These c o n d i t i o n s are i m p o s s i b l e s i n c e the equation, is i n the normal form, Nor can we have & 0 - a and <^c £ o u n l e s s • /> =.^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<x,* ^ / e a , » * ~ 2 a 3** - 57 -~*~ *f <*-i /s tfrt -O ~-w^o~-cS/o * A A6 /or ' 6~ A A6 l0 /= a,/o/?3 /> + /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% +<t**J*, _ ^ / •3 J-7 -t-£—/ •+/Za0 asyb /f - Fa/ — 3 a. 4 /? ~ which w i t h ^ and 2£ - w i l l reduce to + /<£<?,*/= -+/4<?//° d<( which w i t h reduces t o , a l s o t a k e ^ * * ^ where ^ ^ g i v e s = ^ Hence 2^ becomes 6?/ ^^J^-J^/^V&^sj +K*?/° i4*+32*ffy>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; , <?, ^=<p (c) ^ a , (a) 9 = ' as and 3T^ i d e n t i c a l l y s a t i s f i e d . _T~ w i t h c o n d i t i o n s f o r namely 2 4, rf^+as, _ a - 70 -Again i n J • we get J <? 3 g i v e s w i t h S <7 •P i n . J 1 I n 7 w i t h our above c o n d i t i o n s we f i n d - J " i d e n t i c a l l y s a t i s f i e d . ^ T<- J 2 -.2 //sS -h#4§ 7L-S^~^ hence w i l l be I d e n t i c a l l y s a t i s f i e d . Hence / A , / 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 6( £. <? ^y^d-W These c o n d i t i o n s are i m p o s s i b l e , however, s our e q u a t i o n i s i n the normal form. Case 0>). ^ ^ 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 — * ~^***-<S % which i s Impossible s i n c e Hence there i s no i n t e g e r of the form i s maximal under 4 • sa - 75 r where Case;; I I . ? a b le I I . M a x i m a l i n t e g e r s o f t h e End degree i n x When t h e r e i s none o f the f i r s t d e c r e e . I n t e g e r s S ubcase 1, /*>//?, /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 <k •= * #a ; ; vs ^ The work i n d e t e r m i n i n g the maximal reduced i n t e g e r s o f the t h i r d degree i n x when there e x i s t none of lower degree i n 3C , i n v o l v e s , i n a d d i t i o n to Theorem V and i t s C o r o l l a r i e s (see Murdoch's T h e s i s , pages 5, 6 ) , the f o l l o w i n g theorem. Theorem B. I f a maximal reduced i n t e g e r i n fo of the t h i r d degree i n >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 , + <k s? 0 S i n c e there e x i s t s a maximal reduced i n t e g e r o f the t h i r d degree i n then t h 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 of the form R e d u c i n g ^ >( 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 

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