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Determination of bases for certain quartic number fields Murdock, David Carruthers 1933

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. DETEP-MIE&TIOII OF BASES FOR CERIAIli QEARTIG ,-IUlCBER FIELDS  .David. Carru.th.ers Murdoch  A T h e s i s submitted f o r the Degree o f MASTER OS ARTS i n the Department of m_HlMATICS  IKE UITIYERSIIY OF BRITISH COLUMBIA May, 1933.  D e t e r m i n a t i o n of Bases f o r C e r t a i n Q u a r t i c lumber F i e l d s .  Introduction. T h i s t h e s i s d e a l s w i t h the  determination of  bases f o r i n t e g e r s o f the q u a r t i c number f i e l d d e f i n e d by the  equation  ^ +  = ° •  The method used i s t h a t  developed by F . R. W i l s o n i n h i s paper "Integers and B a s i s 1 •• of a Number F i e l d " . The n o t a t i o n i s , i n t h e main, adopted from Wilson»s paper. and  2  The l e t t e r s  ^  and X , Y,  denote a l g e b r a i c i n t e g e r s , w h i l e the r e m a i n i n g  l e t t e r s , Q-, -fr -c,  w  rational integers,^ r a t i o n a l numbers. and  and  (X  and t h e i r c a p i t a l s r e p r e s e n t  s p e c i f y i n g a prims. Greek l e t t e r s denote  The n o t a t i o n a f o r CL  used f o r a . d i v i d e s  does not d i v i d e  We now give a summary of the p r i n c i p a l d e f i n i t i o n s ; and  theorems of Wilson's paper*  g e n e r a l f i e l d of the n**  1  These are s t a t e d f o r the  degree d e f i n e d by the equation  yL + 3^. yc + f  + J3 — 0 0  This- equation, i s assumed t o be i n the normal form, t h a t i s 3  0  ,3,,  e x i s t s no prime  1.  a r e a l l r a t i o n a l i n t e g e r s and there such t h a t  (f^/-B^-si.  f o r ®.ll of Sl~ -1,2-—/"?\.  T r a n s a c t i o n s o f the American Mathematical S o c i e t y , V o l . 2 9 , 1927.  I t i s obtained  from the g e n e r a l  equation  .  --M..= <? d-n  by the t r a n s f o r m a t i o n  = ^  A^'j.-f-/)^.,  where oL i s the  g r e a t e s t r a t i o n a l i n t e g e r such t h a t the r e s u l t i n g c o e f f i c i e n t s , 3o,  3,>~  ~  > 3*T~--/ The  are a l l r a t i o n a l I  integer a  ^ , ^ ,  3  •  integers.  ^  and a l l  l i n e a r combinations of them w i t h r a t i o n a l i n t e g r a l c o e f f i c i e n t s are. c a l l e d o r d i n a r y i n t e g e r s . t  Ot +• ptyc  =  f  denominators of a l l the  > ^) CK  }  or  in  yc .  reduced i n t e g e r and  > 0, ajad i f  ^  if  I f jf  c/^. = -yV  where  i s c a l l e d a maximal  .  theorems of W i l s o n s paper are as f o l l o w s ; 1  If  £  p  Jz  i s the h i g h e s t power o f  H  a  i n the denominator of any i n t e g e r then  i n t e g e r of degree  reduced i n t e g e r .  reduced i n t e g e r of degree >•?»_ i n  Theorem I .  i n which the  _n  i s the g r e a t e s t p o s s i b l e , then  The  ,  ;  I f , moreover, or^r  i s a single-prime  form  are powers of a s i n g l e prime,  then ^- i s c a l l e d a s i n g l e - p r i m e  t  -f- (X^tC  ^ O , i s c a l l e d a single-prime  and ^  ,  i n t e g e r o f the  + —  0  (_<*o j  symbolized by  An  of a s i n g l e - p r i m e  occurring reduced  i s a f a c t o r of the d i s c r i m i n a n t of  the  • f i e l d , equation. C o r o l l a r y 1. in  ^  I f 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  of degree  reduced i n t e g e r i n  in ^  -x  }  o f degree  there a l s o e x i s t s a maximal in  -X  »  C o r o l l a r y 2.  The maximal reduced i n t e g e r s i n a g i v e n  f i e l d are f i n i t e i n number. Theorem I l ( a ) .  A l l i n t e g e r s of the f i e l d can be expressed  as r a t i o n a l l i n e a r homogeneous f u n c t i o n s w i t h r a t i o n a l , i n tegral coefficients,  of o r d i n a r y i n t e g e r s and maximal r e -  duced i n t e g e r s , the l a t t e r c o n s i s t i n g o f ons s e l e c t e d a r b i t r a r i l y from those i n each prime and f o r each degree i n •x. f o r which such e x i s t . Corollary.  The d i f f e r e n c e between two i n t e g e r s i n ^> o f  the same degree  i n yc  and h a v i n g the same h i g h e s t c o e f f i c i e n t  i s e x p r e s s i b l e i n terms o f o r d i n a r y and maximal reduced i n t e g e r s o f lower degree Theorem I l ( b ) .  in ^  I f ^6  .  i s a prime o c c a s i n g i n the denominator  of some c o - o r d i n a t e o f an i n t e g e r , then ( l ) there e x i s t s e x a c t l y one maximal reduced i n t e g e r , Ysl degree  -si i n ix- st >0; (2) i f  in  ^~ -"*"--/  }  ft  t h e r e e x i s t maximal 1 ^>*\-x and (3),  reduced i n t e g e r s o f degrees-^T-/,-A./* f o r each u. }  o < V- <£. t  in. -vc  where  Theorem I l ( c ) .  £4 ^=  0  for Now i f  £  '  f  i  e  g  r  e  e  by o r d i n a r y i n t e g e r s ,  ,^  ,  o f degree  -~~h^} . >w;  in  i  s  maximal  a  -?c  }  then  .<s«c-^»^ i s the lowest degree  in  a maximal reduced i n t e g e r o c c u r s , f o r any K we l e t  o  ^ r j  I f l^j  reduced i n t e g e r i n ^>  J  ^/3 >/3,)  d i f f e r i n g from  YX = ' fro > Y,,  }  there is one and but one  }  s i n g l e - p r i m e reduced i n t e g e r si-  o f lowest  -x.  , st  f o r which H _= -*<--/,  be any s e l e c t i o n of maximal reduced i n t e g e r s of degree in We  -T^T  one f o r each d i s t i n c t prime f o r which such occur.  also l e t  be non-zero  7^..=  •_ '  i  f (j- (, + <  given  ^  *•  ..  and l e t ^  ,^  ,  ---  s o l u t i o n s of  Then i f 2 y- i/  K  s  *^ ' reduced i n t e g e r d e r i v e d e  from  by removing o r d i n a r y i n t e g e r s so t h a t  (y . __ __• • •  •  then the b a s i s of the f i e l d i s  by  It  only remains to determine the complete  maximal reduced i n t e g e r s the f o l l o w i n g  (fk't  '•"  set o f  ^ i s W i l s o n proves  theorems.  Theorem III»  If  A,  A  and  denote the d i s c r i m i n a n t s  of the f i e l d and of the e q u a t i o n d e f i n i n g the f i e l d , r e -  s p e c t i v e l y , and i f Tli , 7°*+,  "75!-/ are the denominators  }  o f the highest c o e f f i c i e n t s o f the elements of the b a s i s as above determined, other than those which are o r d i n a r y integers,  then  A  = f^B^r-  Theorem IV (a) «  - £w It  is: an i n t e g e r and i f El  ^/  p. =  +•  y* +  i s the e l i m i n a n t  and the f i e l d equation, then a l l of  -t of^_ yc l  of  and a l l o f -/L  -  I j X ,  - - --;^-(,  are r a t i o n a l i n t e g e r s ; and c o n v e r s e l y , >  — V  ^ = 0 , 1 , 1 ,  are r a t i o n a l i n t e g e r s , t h e n Theorem I V & )  ^.  i s an i n t e g e r .  F o r the f i e l d  ^ ^ - ^ V - r the maximal reduced i n t e g e r o n l y i f (1)  i f a l l of  0t/  <2. =. O  • y^'and.(37 e i t h e r  0  0  ' jpO ;  J>*^~ U  +B — 0  (2)  can e x i s t ^ V  ;  ff > %  c^l -ft l^r\..  Theorem V .  When a maximal reduced i n t e g e r i n ^  degree  in  ~-*t-/  -pc  degree i n  e x i s t s hut none i n />  -?c  of  o f lower  t h e r e e x i s t s a s i n g l e prime reduced  integer  such t h a t e i t h e r ( l ) , jj-x reduees t o o r d i n a r y i n t e g e r s and a_ ~B^. 3  a l l mod ^  ;  i n t e g e r s and  or (2),  « =  x >  0  £X  0==B*  ,  does not reduce t o o r d i n a r y  3  C o r o l l a r y • 1.  If  —->t-__  i s a f a c t o r of each o f 3  ^  and a maximal reduced i n t e g e r i n A  o f degree  e x i s t s "but none of lower degree i n  yc  then  0  ,3, ,  3^_  t  in  ^>->~f  ^—  i s an  Integer* C o r o l l a r y 2. of  3sx+t  ^v-/  £  0  If ^  ~pc  in  ,  but n o t  but none o f lower degree i n =  -Qjx^  0  x  Corollary 3« a¥ 6  O  CL* +,  If ^ and  7^  then  and ( l ) i f CL^_^ s  = 0  o r (.2) i f a^_-  and  then  ^  OJ  and t h e r e i s a maximal reduced i n t e g e r o f degree  - a, =  mod^  i s a f a c t o r o f 3 ^/}  4 3^.,  mod^  ,  B^-,  then  0 •  i s a f a c t o r of ^O.  S^.-,  hut n o t of _ 5  0  -  6 -  2. The E l i m i n a n t 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 of the e q u a t i o n . .y = <*_  and the g e n e r a l element, of the c o r r e s p o n d i n g  field,  that i s by the product  o f the f o u r  ex, -X  0  +• «  3  -X  A  conjugates 3  J  i  ,  , *x , and 3  7C  -f- Qrx.  f  .+ -efyTj  z  in  where  3  i s g i v e n Dy the norm o f ^.,  1 oC  « -x  r  •x  i  <- X*. +S = o  ^  -t- H-x  3  a r e the r o o t s o f  v  + S  -  Q  T h i s product i s C  EE  <*o »• <  *  -t- *o <x, 1  _  """a <*_.  T<x -ex 0  3  r  0  *  *  Z. ?c  e* c^Zx,  zljl-?t.  Z  0  3 3  3/  yc  + <* «, 2 x, * ^  x  s- ex, -Xf-x^-xs *^  or of, Z rx, '  +  0  3 / I t * .  t  4  3  +• «* of^ Z ^c,  t  +  5  -f- <x o< Z 0  K  <X, of^Z. -x, ^x^y  /  .  J  3  3 2.  ~*3 "X-tf  "f  *o  Z X ,  ^  ^3  «r Z  +•  °<o <*, <*  X  *» <*1  +  3  The c o e f f i c i e n t s o f the <x , -x^ , ^ , and  nomials i n  s  *% o< Z yCf^X ^3  -h <*o  3  Z •X.,' L. H  3  Z -K, ~X-  K  -  a r e symmetric  poly-  and. can t h e r e f o r e be ex  pressed, i n terms of the elementary symmetric  functions,  £•*,.,  -x  £  -x  x  ,  £ ^ ^ - ^ j , and.  and hence i n terms o f Ql , 7?, and  :  S.  h  The r e s u l t s a r e a  follows  o  - 7?  - 1Q -3  n  Z  3  3 3  a  £ y * £ * ,  ^_ 5  = £.  ^  * 5  -  ~^ H  s O  41  i.  Z  4  i  Z  i_  rc^ -7C  x  3 A  ^j-x^  =  Q S  ?«j * v  =  - TPS  £ */ /  J  =  +%S  3  J  Z  = -KS.S  3  —  ^ t*^  z  a  *• 4  Z  3  3  Z  t  /  J  7c,  ^  ?<  L  *j  ^1  2_  =  s  =  0  ••/•••  4- 4.  Z - .x, *  4  J  3 7C  3  -x^  x  =  a s  ^ Z.  x  y<  .*  2L  v  ¥  v x i - x , •x'j  Z  X K  =• S  3  ,  -X*  +-($/  ? 3 R  + 5 <*,  -  X  -  i-Q. tx/cx^ — 3 %*oc<  y  - 3 Q S ) t * / -  L&S  ft  cx „/ 0  cx, or •+ Q.S cx, cx 3  %  I-Quocx?  —  3  —  -  3  £  +  . y e s  —  3  "PS-i X  we get  (Q +is)<x**l x  + (R -- ±&s)cx <y/ 1  1 -*  4 3  2,  «V  a n  =r - h S  S u b s t i t u t i n g these i n the e x p r e s s i o n f o r •==••  -  2  . -  £ ^/  E  •vc  t. • s.  3c ^ 3  2-  - *s  0  +{GlS +S-S JcX,oc  3  + (tQKS-tf)*,  -US  <x,cx  3  x  j/  <y  3  - 9 -  (hQl S  - 3  •  .* *J *<>  _fow f o r the f i e l d we have  7?=o  * <3 ^ * s - o  and the e l i m i n a n t reduces t o 3  ^- 6 o',  — 1 AS<*V cx f- QS <*/ cx +• (QS't-Sj 3  + S**t-•+ •Q.s''txlot3-- + S cx 3  3  - / g o< oc^a l  /  /  +.(us•-%<£)ce?.oe, #2  / (u S^- J- & S) <x X  3  ~ l&S c<, cx^  u  0  MS«,*,*••«±  t tffiS* of, * * . 0  %  3  T h i s r e s u l t has been checked by S y l v e s t e r ' s method o f elimination.  We a l s o  "a  have  2-\  dcx^dcx,  By Theorem 17(a) ( i n t r o d u c t i o n ) , i f  = #o + <y, <x + & -x- +• off ? %  i s an i n t e g e r then a l l the above expressions i n t e g e r s , and, c o n v e r s e l y , are r a t i o n a l i n t e g e r s then  are r a t i o n a l  i f ^77 - ^ r ^ ^c-  = > 0  +• S — o  >%) 3,  i s an i n t e g e r .  To f i n d the d i s c r i m i n a n t o f the f i e l d 7c +ft^  1  equation  we use the f a c t that the d i s c r i m i n a n "  o f any q u a r t i e i s equal t o t h a t o f i t s r e s o l v e n t  cubic.  In t h i s case the r e s o l v e n t c u b i c i s Q. \t *" - h S <t  ~  7':-•*>/.  -/-uGiSeZkO  or  The three r o o t s a r e the  discriminant,  or-•  A  Q  ,  Vms  ,  and  , i s g i v e n by  —Yus  and hence  - 11 -  2.  ReQapitulation« J[ ~ cs +. of, -j* 4- «^  If of the f i e l d .  3 6  a l l rational ly  =  + cx -pc  0  *•  i s an i n t e g e r  3  4s = o ,  then the f o l l o w i n g are  integers. cx_ +M* tx,  - i.  +(0- tzS/iXo  e  <<,-£. a$ <x, p< + Q.S, <x, « 3  x  « _ •+\@.--3GLSJ<x oc - ,2 6 S «•„ 0  3  +(Q $ +1$ Jo., or - SlCS of, ^ }  y. 5 «*_  3  3  \^=.  J  k 6 c l *  ' =_  I fe)=aL 5  \  x  L(a t-%s)x (a -3qs)#«<* -ias  + £&<xca>*>  L  fi  -5'fis;^  3  It I, , 1  3  t 5 J #i +. (a - 2 as) 3  , I  3 t  and 7^  are r a t i o n a l  integers,  i s an i n t e g e r o f the f i e l d . The  discriminant  ^  <^^.A',  *(*.s-a J 4  1  3  0  - ^ficf. ^_ * c  Furthermore then  0  o f the f i e l d , e q u a t i o n i s  From theorem I, i f  ^?  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 of an i n t e g e r , then  £  (<a .  The work of d e t e r m i n i n g the maximal reduced Integers 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 cases as f o l l o w s : Case I;  To determine the maximal reduced i n t e g e r ^ o f the 1st  Case I I ;  degree i n  To determine the maximal reduced i n t e g e r s of the 2nd degree i n degree  X  when none o f t h e 1 s t  exist.  Case I I I ; To determine the maximal reduced i n t e g e r s o f the 3rd degree i n  when none o f lower degree  exist. Case IV;  To determine the maximal reduced i n t e g e r s of the 3rd degree i n x when those o f the 2nd degree e x i s t but t h a t of the 1 s t degree does n o t .  Case V;  To determine the maximal reduced i n t e g e r s of the 2nd and 3 r d degrees i n Y whenone o f t h e 1st degree  exists.  - 13 -  3»  Case I* I f there i s a maximal reduced i n t e g e r o f the  degree i n " We and  a  D  ~ o  f o r i f so,  on s u b s t i t u t i n g t h i s i n ^  we get  £ = (~^T<, '\  i t i s o f the form  cannot have  10.  and  1st  1  y =  and  3  'jS  ^t, ^  respectively,  which i s i m p o s s i b l e  since  the f i e l d e q u a t i o n i s i n the normal form (see i n t r o d u c t i o n ) . Moreover, ^.  t  t  o  Theorem I l ( c )  ~  £  and hence we must have  0  From Theorem I V ^ ) , s i n c e  ^> /zQ. tf  (-JT  whence  > f )  where  If  a-bove, —i:y^_K and  fi  ,  Substituting /this i n  I,  i:*  are s a t i s f i e d , /  + ~*  ^ =X  we have  £  t, — t  b  j %  t _<__ z. and <j, reduc  and  0  t £ i_  on s u b s t i t u t i n g i n  Therefore ^  fied identic/ally,  o ,  0  By  1$  we get  -b-at ftbj.3 i^--^g^s^ih±e^-^iTr(5ei—fre-m  We have  0  a  t = X  Q == - 6 mod, ^  and  then  0  and t h i s has been shown t o be i m p o s s i b l e .  0  to  t, > t  l e a d s to a s i n g l e - p r i m e reduced i n t e g e r f o r which  a. —O  and  for i f  ?=/  and s i n c e ~ X ^  Gt> - /  and  JT, --- ?.--  i " ^ and  — _Tj  S =E —(& v-/),mod /6 .  ^  F  X  becomes ^ i  s  H  ^ ; j)  satis-  Q. = 4., mod ^  if  I f these two  conditions  t h e r e f o r e , we have the maximal reduced i n t e g e r o  4,  Case II»  Theorem A.  f i r s t prove the f o l l o w i n g :  I f a maximal reduced i n t e g e r i n  X  degree  We  in  ^  of  e x i s t s but none o f lower degree i n  then 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 jz~(a G-,,0  such t h a t  O}  Since there e x i s t s a maximal reduced i n t e g e r i n ^> of degree X  in  ,  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 of the form  >t a  j 0- Since  ^  i s an  i n t e g e r , so a l s o i s jf-x - R e d u c i n g t h i s by means of the f i e l d e q u a t i o n we have  The d i f f e r e n c e  i s a l s o an i n t e g e r . and rxr  C i '~ a i  °  a  The d i f f e r e n c e between  + )JJQ  a  ' jf  i s an i n t e g e r of the 1 s t  ~~ J^ degree i n  and must t h e r e f o r e reduce to o r d i n a r y i n t e g e r s s i n c e ,  by the h y p o t h e s i s of Case I I , there i s no s i n g l e - p r i m e r e duced i n t e g e r of the 1 s t congruences  degree i n -x.  From t h i s f a c t the  •  f o l l o w at  once. Y/e now  d i v i d e Case I I i n t o f o u r A*  and  B.  and  G»  but  _Q«  Sub-case At- //&  and  f/S  but-;  //a  4/S  sub-cases;,  »  I f a maximal reduced i n t e g e r e x i s t s , by Theorem A there  i s a single-prime  reduced i n t e g e r  ^ x  (&o > Q-, > l) such  that  or,  since  ^>/&  and  /S,  •6, (a/^CLi,)^ a a;-J.  ^  0  From  (I), either  both.  I n any  £  = O  0  case ( 2 )  gives  reduced i n t e g e r e x i s t s , there integer and  or £  e  =_ £<,,moa.^ = a, —0.  7j_ ., 7  0  , and  1  H  are  Hence i f a maximal  e x i s t s a single-prime  ~7~ •• ' S u b s t i t u t i n g t h i s i n  or  I,  s a t i s f i e d by the  we  get  reduced d>  ••  same c o n d i t i o n .  — 16 «••  Hence i f  £IS  and. if  ^  there  i s not maximal,  a  0  gives  = 0  /__>  I  and  ~#  1  <x,~o  and  1  ^ ) •  cannot  fi /Q. X  d  if  —  It}  ^ 9^ £  Ii,  and  c  &, ¥ O  and  0  /' Q-  gives  we have  I,  which i s im-  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 _Jor can we have  there  f o r i n t h i s case gives  3  '  "by Theorem 11(h),  must "be an i n t e g e r o f the form have "both  i s an i n t e g e r  ^ =-5.  unless  T  whence from  t  •jfi*/9 .  form.  Hence i f  for,  3  ^ ^ i.  the  o n l y p o s s i b l e i n t e g e r s o f t h i s form are  Substituting mod y ^ and  1  or £6. fi= 0  we f i n d  mod^^  whence  ^  in  -Z"^  + &'f~. y  Gl r  "  )  and  i £ ^? =  £^  0  I  Substituting this i n  Q. - h S = Q mod^ Q.  we get  and  ^ /S  Q.*- 4-S  s  = £a.,y£\  which i s i m p o s s i b l e .  ¥  There can, t h e r e f o r e , be no i n t e g e r o f the form  Hence, i f the  2nd degree i n jf?  by Theorem I l ( b ) there .q ^ o  where  the o n l y p o s s i b l e i n t e g e r of i s (~  }  o  }  ^y,^  I f t h i s i s not maximal  i s an i n t e g e r of the f orm  t < -^-f  that t h i s cannot e x i s t unless  and  ^  _= 3.  0-, = O -  We  ~p  J  s h a l l show  -  From hence, s i n c e  I  ^0  0  we g e t , assuming t-/  ± & ft = ^  '  From  _2, ^ o  that  /  moi^'^  e  ^  T^Ca) ,  /d  ./"j on s u b s t i t u t i n g  Q - U S = and  we  af'a  We f i n d a l s o t h a t X  and #5 = £L mod ^ ,  a, = o  and j T  3  ,  ,  w  e  get  4S - f>  ®  i s s a t i s f i e d by the same when we put  r  S-  -fi^s'  we g e t  i s impossible  f o r &, ^ O.  and the only p o s s i b l e  Substituting this i n mod ^  }  ^  i n view o f (3) t h i s  Therefore  however,  find  -Z*  From  Q. +  and on w r i t i n g  £2 —^ko?  condition.  =  0  Oj mod ^6  i.  3  la jf>  and  and t h e r e f o r e  and our i n t e g e r becomes  From  But  -  we have  u  a  17  I  ;  3 ,3 • , „ i /1  we g e t i . A o ^ = = £1  H  .2\ and I,give  , _  integer i s  the congruences:  i  , j  - 18  xa f> = fl ^ -f>™4  Substituting  that  Q -:#Ss  mod ^  consider  the  (a) cio - o  a  (c) (a)  CL.stl  t h e r e f o r e , three  Qlq-  o  and  the  (b)  and  a, ^ o  hence  ^ .  If  is  &-  o  0  ? -pQ  and  possibilities  a  ^ o  0  a, ^ o  and  Since ™ _ ^  .  i n t e g e r i s of the  cannot e x i s t , however, f o r and  mod  a., •=. o .  and  0  ^ = 2_ .  case  a,  c  cannot have both  (X, ^ o  and  /^ ^6 =  i s an i n t e g e r o f the form ( f f •> ~  shown above that we There are,  and  __ 1 , and  ,  not maximal then there I t was  .  --n. i s the g r e a t e s t i n t e g e r such  where  now  are  have the maximal reduced i n t e g e r  > ^J  We  f i n d t h a t a l l three  c o n d i t i o n GL- H S= O, mod ^  Hence we '  we  B  s a t i s f i e d by the  \^-i  -  J  f  x  _= x  w e  (o, 4r , -k)  form  bave This  g i v e s ^ ^ r * yi aaad-—£  requires that  This i s  _/  / *">  impossible  s i n c e the e q u a t i o n i s i n the normal form. (b) and  a  0  *f o and  ^ o , _.  the i n t e g e r becomes  this in and  I  Z, are  x  while  —  I,  _T  ¥  (j. ' T , we  I n t h i s case ' "I }  satisfied i f  requires that  1 /ft  Z  y  =• / and  i s satisfied, S =_ 4>  and and  0  -/  Substituting  1  f i n d that  a  S" ==  mod  _T  3  mod /6.  If  these l a t t e r c o n d i t i o n s are s a t i s f i e d , therefore^, there i s an integer  > -£ >  ,  - 1?  -  i f t h i s i s not maximal there i s an {~£ > -75 \j}  3  ' 2? i -  T h i s i s the  "T ' i v  only p o s s i b l e form, f o r  would, l e a d to an i n t e g e r  which, s u b t r a c t e d  from  (•£ > _  1  ~b-) ,  which cannot e x i s t , w h i l e (•£ > (0,  integer  ~%_ > f O-o  sidering  ^ = ("J»  1*1 Q. and ^  i m p l i e s the  have X  5=  0- =  , mod  /6  existence  of  and  0  _~j  f i n d that  0. —j I 0  }  X  3  •Q;'—  / mod  gives  =  Writing  G i ' V O ,  (%.  A = ?a'  ,  If t  a s  0  M  ?  we If  I,  and  impossible,  / mod 7/  1, gives^p/f-/)  ;  --/  giving ,  _T,  c mod  >  /^-./y,  cannot e x i s t ,  have the maximal reduced i n t e g e r (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  being  if)  and  >  J  Q. = O mod. 8" t  and  = ••tf mod /_,... i  (c) form  a  0  ^0  and &, —Q > o  }  9  "J-^J  I f there we  shown above t h a t (-£ > o, -jy.) in  i s odd.  hence  Hence the i n t e g e r >  5  -^e  but t h i s i s  ;  I  s i n c e -4. i s odd.  which i s a l s o i m p o s s i b l e  t h e r e f o r e we  of  also  S = /6-4 + H  6>V-^= _),mod  mod. /6  and.  }  We  and  which i s Impossible. mod /+  existence  > iO  3  i s odd and  #  con-  S u b s t i t u t i n g i n I,,  requires that  x  In  the  conditions  s i n c e the  }  i s s a t i s f i e d provided  requires that f o r from  would, l e a d to  s t i l l have the  <2, == -/-  is satisfied.  H  ^ ,  .; '  we  }  > ~^-J  y 0  d i s c a r d e d i n (a) •  I \  > ~*f ~p)  (Ji  g i v e s an i n t e g e r  >  which was CL,  integer  X„ , —  ~r,I, ,  we  get  i s an i n t e g e r  must have  ->-» > t  of the as i t was  cannot e x i s t . S u b s t i t u t i n g  if. O-c S ' Q_.-^ywc£ X H- Q-o ~ /*.*<>&+Gl + s-S =  3  76  i  n  +« &  ^"-./SaU+^a+ta's  a \ n  c  - H a  -  From ( I ) we have  e  0  $ - ^  (I)  O ^ ^ d  tL  = 0 ' ^ *  s  a s , S  x  (  (Z)  "  X  = O ^ ^ U  Q + £~ 'a .  - " O )  .  --(*)  Substituting this i n  (2) we get ja.  'X  — Q.  + hS  = Oswvd  X  and. t h e r e f o r e  We now l e t  US  -  Q.  X  + i * *  %  4- ,  ana s u b s t i t u t i n g we get = Oyyuuttl  30.^^4-  8  (S)  S i m i l a r l y , s u b s t i t u t i n g i n ( 3 ) we get  and. from (4) 4.  From ( 3 ) we have  4- -  == Oyy^K>  $•<• - 3&*'.  ^  Substituting  t h i s i n ( 6 ) we get A<-a.'*_ 0 mods', and t h e r e f o r e and  if  a. = % of  we have •<2/_= ^c, mod 24  X /0.  - 21 -  Hence, a.' and aJ  and  mod  ,  -e  -c  are e i t h e r both odd or both even.  are even, we have  and t h e r e f o r e  a- = o, mod  = 0, mod /6 ,  and  If  and £a*V-^___>  and  ft^**^^  But i f a' and  are odd we have from (i>)  / 5- a,' -e- -6- = _) s^yMn&( %  ~£ 0 siwfv&L /6  1  and  therefore  Let  -<£ = A-^  where  a.  and t h e r e f o r e s i n c e  }  i s odd, then  =  / mod >*/ ,  we have 4- =.-j  T h e r e f o r e i f q\ us = l*(n<£-0 j •& maximal reduced i n t e g e r  5  °> "^J  of t h i s form, t h e n  there i s a  where  b u t =j£ Q., mod 5- .  ao == Q., mod 2.  If  mod # -  ^  =  G -#S  and i s not  i s the g r e a t e s t i n t e g e r such that  . it--*, f a. /• >  JL  /(Q. -n-s)  Sub-case B:  and  Q.,mod£.  ,£ j>& and,^ ^jy.  From Theorem A, i f there integer i n ^  .  o f degree. • X-  ( a, -&o x  0  in  ^ =j(cLo , a,  prime reduced i n t e g e r a  i s a maximal reduced  *-«.)  = S  J  s*r , there i s a s i n g l e }  /) ^v^rd  such that f>  -~-Q)  V 2- ^ t ' /  .  y  /  2-  - 22 -  •• •CL;(CL, 'V  == a a,  G. +-QJ  i  1'.  o  0  o mod ^6 , and. h e n c e s i n c e  From l^-fc,)  CL, — 0  }  Therefore,  from ( l ) , s i n c e  we must have  and the only p o s s i b l e i n t e g e r i s o f the form where  O-^^O.  o f the form 3-s@-)  get  ^ —  '  mod ^  0  this i n X  3  '•• - T ' and  Fc ^ ~  ''  t  O- -  = }  =. o ,  -^J  }  such t h a t  ££  i f ^> ^ 1.  > o'  integer  0  ^  1  ft J  b e i n g zero from %. . Then from  — ft •+ ft 4i .  US  mod  we  I  H  Substituting  ^  there  where =  i s a maximal reduced ^  i s the g r e a t e s t  •••• A^n  o,  mod ^  ^  and  Xa =. 0  Q  mod ^  the o n l y p o s s i b l e i n t e g e r i s / o f the - Substituting i n  T h i s can only be s a t i s f i e d i f pos s i b l e  ( "pr '  JT, , we f i n d that they a r e a l l s a t i s f i e d  Gl~-u-S  Therefore  Integer  or  }  by the c o n d i t i o n  form  •/ •£/  0 J  ;  ^a =  c  A maximal reduced i n t e g e r w i l l t h e r e f o r e be  C o n s i d e r , f i r s t the case  •  a ^O  i n t e g e r i s (-^ > O ,  get  which i s i m p o s s i b l e no maximal reuuced i n t e g e r i f  X y^e H  get  - 23 -  Sub-case G;  £ Is  but ^o^Q. *  A g a i n , from Theorem A, i f there i s a maximal reduced i n t e g e r i n  jf> of the 2nd degree  there i s a s i n g l e - p r i m e  nc  then  ^ — ^C^o,  reduced i n t e g e r  j 0  i  such t h a t  Q-c C°- #l -  Since  in  ^/S  s  S , W ^  (7J  ( l ) becomes =  CLo(o-r t-O-o •+-Q.)  = 0, mod ^  From -/-fa.)  ;  we must have  a,-0  and t h e r e f o r e  either  We  D ^iwd  ' (X-)  ^  . Hence from ( 2 ) , d (a-^ )= 0  a  Q.  and t h e r e f o r e , s i n c e  - 0  c  cannot have  a  0  the i n t e g e r becomes  or  }  0,mo&^  0  <2 = 0  — O  ^ mod ;  f o r i n this  and s u b s t i t u t i n g i n  I  3  case we  get . Q  -f- X S  which i s i m p o s s i b l e s i n c e possibility left, ft-o s  From X  d  y  v  ~  0 sww&t  (f> / S  therefore, i s  but (~  * °  mod jft  , //fl 0  o,mod ^ XGl=.  and t h e r e f o r e  O , mod  $ $ & J  The  only where  <— ^ . /?  /  /  _.' •  ,:  /  <^<?  <<-/  ^  /  /  ?iu-c*~*-  - 24 -  Sub-case D;  /g  but & j>£  In t h i s case, of the 2nd degree i n reduced  i f there i s a maximal reduced, i n t e g e r  -?c  then there i s a s i n g l e - p r i m e  i n t e g e r —jr(&o » °-t •> 0  and s i n c e  (ft/Q.  the con-  gruences o f Theorem A reduce to a. (&,  ~. S <m^/^>  — o) a  1  a  ------------  a, .(a; " — #o) = a ay^ym^/^  ---~  3  0  From ( l ) w e cannot have  a. -o  -• (%•)  jj£.  The o n l y  pos s i b l e cases are t h e r e f o r e (a) cx ^ 0 , CL,  O and (b) a  7  0  since  (j)  0  a, — O .  Also  and t h e r e f o r e , s i n c e  from Theorem I , j,(a  but  rf>Js  Hence the o n l y p o s s i b l e i n t e g e r s a r e  Substituting  ^,  =£  0  0  /£S (Q we must have r =  we get, from  > •£ ' f_)  J  3  ^ = jL • and  -  whence and  -  therefore since  S  x(i-s)^^/h-  =  Q  I  i s odd we have k /Q. , From  x  ft.  k ~ H&. +  or  But  t h i s i s impossible  no i n t e g e r  since  and hence there  can be  (jx ' I ' x) . We now s u b s t i t u t e  are  == Oyyuod%  s a t i s f i e d while  J.,  u  i n _T, ,  J  —  .  u  I  I. I.  gives  / — xq +•0*'f % S - zas + s — 0,^uxl 16 r  (7  or  ^-1^3  ; a = O, mod ^  5 =_  mod 4* .  t h i s condition i s equivalent to By the hypothesis  no maximal reduced i n t e g e r of the 1 s t t h e r e f o r e the c o n d i t i o n s mod  ,  == Ql—I,  o f Case I I there i s degree i n  a = i . , mod u-  and  and 5=  are not s a t i s f i e d . Hence we have the i n t e g e r  S  - 0 *^u®c( [f  -5 =  whence  Since  O y^ad /6  mod # ,  and p r o v i d e d  ' > x) 0  the c o n d i t i o n s  provided  are n o t s a t i s f i e d . ' °> ~k )  If  an i n t e g e r of the form However  i s not maximal there must be > ~p '  (  J  t <c  where  before 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  s  necessary  to f i n d  the c o n d i t i o n s f o r the e x i s t e n c e of the  integer  J  ' 0 > J^)  =. (j^  implies that o f  Since the e x i s t e n c e  (j_ > & > f )  77 = ji 0  w  e  bave  &  5 = Gl.-I  Q.-J -f-u-4  Also  since  = -^7  0  We f i r s t dispose on w r i t i n g  £  £ = (Q.-/),mod H  we have S ~  and we may t h e r e f o r e w r i t e  of  o f the case  £  e  = -/ .  From  _Tj ^,  we get  Hence, i f Cl = 7-Q. Q_  Therefore  Ql  .  — '  /c  S u b s t i t u t i n g i n 7~  z  = 0 ^i^ati  if  Cl ~ i.  i s odd, and  ana  the l a t t e r  4- I  u .  and- 7", the former i s s a t i s f i e d , ;  -f- A  +. 4.-S  S  = Oyyy^Ci ix  whence  while  gives I  Or  (3)  •  (/  * A+S) 5  & Qstousd %  ft)  But  (3) and. ( 4 ) a r e the c o n d i t i o n s f o r the e x i s t e n c e  maximal reduced i n t e g e r o f the 1 s t degree i n by the hypothesis integer  Jf  o f Case I I , they cannot be s a t i s f i e d .  L  Q  do — I  (Ol-x) mod  t  0, /6  We may now w r i t e that  J. _  we have from  mod %  }  S =  GL- /  or  $ a n d X  >••.  s u b s t i t u t i n g we  %  v  4,  ._  = O^p^cii^  5 therefore  /6  we have '^W^L g r e a t e r than integer  %  f6  being  Q — i . , mod //. and }  f  We now c o n s i d e r  t < ^  Z  have the i n t e g e r (jz > & > J*-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 mod  find  -f-Q ~o?w*d 1  whence  We  and  gives  + S)  (/ ~ a  the  s  Eence we have 6=_ ,2,mod ti  / - lfl^ci^'is-jfts  where  I  and s i n c e i t i s a p e r f e c t square,  i s s a t i s f i e d , while  S = 6a  The  0  _Tow i f .  (_»-2_)' =  and hence,  therefore., cannot e x i s t i f CL = - /  t  Therefore  of a  the g e n e r a l  integer  F i r s t we take the case Hence i f ^ -1 s i n c e then fJ- , r> -L\  X  ^ 0.  1 , ^  £  From cannot be  z  would give r i s e to - ^ j *  f o r which we must have  /  Q. ^  X  r  mod. H-  and. t h e r e f o r e since  be g r e a t e r t h a n ^  X I£t  t_  cannot  f-.sM.  2-.  I f ^ - z * =/ which i s the o n l y other p o s s i b i l i t y , • • •• g i v e s r i s e to a n i n t e g e r j , ~ (~r •> x > , We s h a l l }  t-r 1  ' ^.  now prove t h a t t h i s  cannot e x i s t .  i m p l i e s t h a t o f (j.  o  }  ~)  }  and hence we have 5 = -(6LH)mo&. tt  and we may w r i t e £ = uiz ~Q.-/ we have  o-o — I.  A l s o s i n c e - £ < ^° __ •£  . substituting i n  therefore  write  .  a<, — */. If  and  The e x i s t e n c e of  -S = %  mod - Ql-I  ,  •••/ *  +  and  we get  3  We may t h e r e f o r e  and s u b s t i t u t i n g we f i n d t h a t  J,  i s s a t i s f i e d , while  JT  T  x  gives  -f-H-S-f- 2. Q.S +S'<-.•—. Oy^uw( X .  5 _=  whence T h i s i s impossible  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'  reduced i n t e g e r o f the f i r s t Now i f or i f « = i «  a. -=.-i 0  }  degree i n ~xr.  from  I  3  we g e t  /  Cl* +• i + x-A == o^pt^d  \  l  t  yk^Q c^-fJU^  CMM  iAe^  n-t~1  Therefore mod h ,  &' i s odd and and  ^  i s odd.  Q  =  Henoe Q.s 2.  / mod ^ J  Substituting  in  I*.  and  I,  the former i s s a t i s f i e d , while the l a t t e r g i v e s  5 = U-k  Substituting  and  Q =ift'  -^4—^bA^-is-i^^eeibie-j—fori—since—^—and: P^£^  C  y J t  e-dd-i—every- term—esc-gpi;—  ^^^~ " ^" V  rs—divisible  ^—-are—be^tb by—  Irenes there can bo no ini-^ge-r Tho-?ofore v;c cannot have  ^-=-f-  -ahgr?-&®-~^teow& abOV-e^ •  becomes  If  '  ^  '  mod  - The I n t e g e r  7  ^  then t  >  ^-t  Si L  therefore  f. = (-ji >  0  T h e r e f o r e , since an i n t e g e r 0  J ^ ) - __  1  / ^  i s an i n t e g e r  e x i s t s , but  > ;  and t h e r e f o r e  /a  i s a l s o an i n t e g e r .  we get  'j})  > If) > o , -AJ  does n o t , we must have G -~ J 0  . Substituting  ^  in  -Z^  we get  // Co ~ i. A = O svwiHX X and t h e r e f o r e I  3  we  Therefore and,  ia  0  =  '•n-f  a + X""a  .  On s u b s t i t u t i n g t h i s i n  find  £  1(0.*-us)  and we may w r i t e  s u b s t i t u t i n g t h i s we get  Q?~-ifS =  ZT'^  A\  Therefore  since  Q = j  mod 4  Lj  we  have  (s)  Similarly,  substituting i n  and  ~t  Gl-hs  we  1^  from  (A *  l  - ^ =  This  is  Z ; mod  impossible,  Therefore H  e  n  G  a  if  and  therefore,  is  a-ls  odd,  for  from (7)  l&o  Q. mod # ,  (6J  (  therefore  0"-^5  since  2 o W/^  and  and  £•<> S  a.  must he e v e n ,  e  Therefore,  ^.  - + X  + 8-£a  F r o m (j>)» t  =.&+£.  J.,  -4-)  x  XO-c  get  a f a ' ^ a - ^ j and a g a i n ,  and w r i t i n g  ^ But  =  it  •£&•-•/  since  a n d from  O ^ W l  ~z&,/yvurii  if was  *  we l e t  A  ^  -"""(7)  mod 4  and hence  5  ( 3 ) , 1  mod ^  "  £.  a  = X-O!  shown above  that  we a  have 0  = /,mcd #  -  31  '  a  Since  o r i f we  l e t CL = £  -  _= 3  tf/,  _  -4-  -  congruence (7) now  a~£a^a.nd  (V W - ^ ' J low It  f  ~  becomes  u4-'  0;swd/(,  S  i s even  -  #J  0^ mod if and t h e r e -  f o r e , s i n c e i t i s a p e r f e c t square  a'  and t h e r e f o r e But  x  t h i s i s impossible  '  since  + &' S a ' ==  since  4  -  Therefore  i s odd  + a'-•£').&..  Since  Q  hence  ft--£'^0  e  mod A .  •"  cannot be even. From (9),  w  ^ ^ / ^  h  a  v  and  0j-wrf-i.  are b o t h odd, mod^  a  must be even and  and s i n c e , from ( 8 ) ,  a ' == / mod ;  e  and we may w r i t e  ' = tf-l — l  and s i n c e  #/  i s even,  ^  Hence i f C - ^ S = . 4 . T V mod. $  then there  •••) j  •»  ±.  J_a  = fi mod  6  only ease l e f t  As b e f o r e ,  f  =  since  i  '  (%  1  to c o n s i d e r  that i s -when there  .  ^'^and i f Q~  e x i s t s a maximal reduced i n t e g e r  / where  ^ the  ^  a  i s when  t  - o and  i s an i n t e g e r o f the form  °  ' i^.y  ^  ° ? ^J  Z  ^  =  ^  we have  >  J  L  a ~ /  we must have  6  mod l+.  therefore  ;t£ =• Cz *• i.^  <X ,  6  § t S  3-X.  /-n-i i O. -  i.K--!L ly  2-  Therefore  = Q., mod £  0  . • and  Substituting this i n  S . Ojsnwoi  + US  '^SJ  1  3  we  X-n + l  i  t  .  %  I(&  £a  3.-*.-% /)  ^ a n x i  w  e  may w r i t e  Q.-US  — J-  whence the above congruence becomes 3 ^  -4-  Substituting similarly i n J  and  from J ,  Oy^>^§'  (/O)  we get the c o n d i t i o n  ;  4 - a  From (10) we have  (11) we get  ^  ^  s= ^  .-. = -.o^yyv^d/6  + J CL  %  --  and s u b s t i t u t i n g i n  "00  /  ^  Is  It  2* y  J^L^^-e^.  (v  •••  (7J  1  I^ju-ijl-  JL-ya-A-'£r *  i^^Z—tj^^  y)  - 33 -  Hence  a.  Therefore  a.' and  I f G-' and  A  are both even, U I cl and from ( l l ) ,  a ' and  Q_ ~fys  •.•=•./_,  ^  we  GL -4 S  have =  X  = • • m o d 44 ,  \tf-<t+0  S  we  and t h e r e f o r e , s i n c e  = / mod #  Q =.  Hence i f  mod g',  i s a maximal reduced i n t e g e r  i.ao =. Q.,  mod x. ,  but  ^  i s n o t of t h i s form, there where  i£  0  =  amoii. #/-t£ but  and  m  since  t h i s case,  4<2 == &, mod  £ „  0  ^mod  In both cases, s i n c e  there  and  t  can t h e r e f o r e w r i t e 4-- *f4 and t h e r e f o r e  mod ^  therefore, 0  ^= o mod 4. •  /6  are both odd, we have from ( l l ) ,  and we  Z£  whence  are e i t h e r both odd or both even.  Hence xn t h i s case If  a = 2. a\  i s even and we may w r i t e  ^  i s the g r e a t e s t  However, i f  5 ^[ mod  have  >  Q. =  ,  and i f  ^  2 , mod  a  ^0*^*0,  =  ^~Sv J 0 .> J"-*}  where  1  <3_, mod j i  I f fl - ^ 5 .  .  i s an i n t e g e r (~~^ > i n t e g e r such that  GL~-h$~ 1*''** (^u<4.-f)  0  X  L  }  > jf^J /09- "  the l a t t e r i n t e g e r i s  not maximal since i t has been shown t h a t i n t h i s case /Go  i s an i n t e g e r / ~j~+t  }  j_  %_  }  j_  \  ] •  T h i s completes. Case I I . summarizes the r e s u l t s ,  there  The f o l l o w i n g t a b l e  that i s , i t g i v e s a l l the maximal  reduced i n t e g e r s of the 2nd degree which can occur when there  i s none of lower degree, together w i t h the  necessary f o r t h e i r  existence.  conditions  -  34 -  TABLE Maximal Reduced. I n t e g e r s # rL. Sec^Uh^ywsL""  I.  Conditions f o r Existence  Sub-case A; £lQ.  and//s  1  —  //a  . a n d / V - S , ( 2 ) , ( 3 ) , and ( 4 ) unsatisfied.  ; 2.  i i . V X  ^  ;  =- O,mod > ^1. >: 4. ^ i s the g r e a t e s t i n t e g e r f o r which ^i"^ * t h i s congruence h o l d s and 4 a ^ = a.mod^"", ^ ^ . s  0  0  &  4.  ^  1  +  i-/ft  and S s  andzYs,  mod. /6 , #9 u ^ ^ ^ ^ > l .  a = />smod£  , ^ > ± - If  a * - * s i s of the f o r m ^ ( W - $ > , ^ = a n d ^ . s a mod X", b u t " ^ Q, modX^*'. But i f e - « i s n o t o f t h i s form, then -m. i s the g r e a t e s t i n t e g e r such that,•.•£*'"**•/<&*'and = Q, mod i ^ f x  Sub-case B;  g r e a t e s t i n t e g e r f o r which t h i s h o l d s and za„= <2, mod ft™ Sub-case C: ^ / S but ^  7  '  the congruence  There can be no maximal reduced i n t e g e r of the 2nd degree i n t h i s case.  -  Maximal Reduced I n t e g e r s ^~pL5^^ Sub-case D  33 -  J>^y^--  Conditions  for  Existence  t  j / q but 36^,5 G=(s*/),mod^ ( 7), ( 8 k (-9) and (lo) u n s a t i s f i e d , and the c o n d i t i o n s mod*, and s = mod/6 not both satisfied. •  :  • /.••  •• •.  / " " " ' ;  :  & == £ > m o d a n d s == (a-/j mod/6, ( 8 ) , (9) and (10) u n s a t i s f i e d Q = l  8«  mod & , q^-hs - x'(n-t-i)  }  —  +• i -x +-h,  and  }  ZG-o = C, mod z'  Q=l,mod 8-. ,  I f G. ~»S=l  0  mod A.", but ^ Q mod + , •>*.. i s the 1  g r e a t e s t i n t e g e r such t h a t and o.fl == a mod %T*! ( ^ a  0  G = i,mod 8,  2*- Z. £/• J  i. ""**-1(0?- »$) to) 7  a - # 5 = i _ ~ * ' Y ^ * ~ ) and  j. a _= 6,mod £ . 0  (ttJ+t), sn>i,  x  and zo. =q I f ft— ks =  y  , svx > i  - 36 -  3•  Case I I I ,  We s h a l l now f i n d the maximal-reduced  i n t e g e r s o f the 3 r d degree i n of lower degree i n  ^ •  when there  e x i s t none  The work i s d i v i d e d i n t o the same  f o u r sub-cases as Case I I .  I n a d d i t i o n t o Theorem ¥ and  i t s C o r o l l a r i e s , we r e q u i r e the f o l l o w i n g : Theorem B»  I f a maximal reduced i n t e g e r i n  3 r d degree i n then there  -f  o f the  e x i s t s , but none o f lower degree i n  e x i s t s a single-prime  reduced i n t e g e r  such t h a t  al-  Since there  a,  = -Q j  •  e x i s t s a maximal reduced i n t e g e r i n $  of the 3 r d degree i n -as then there  exists a  single-prime  reduced i n t e g e r o f the form  Reducing  ^  by the f i e l d  f  t  S u b t r a c t i n g t h i s from  o  Since  o  f  equation,  P a  *£  we have  f  ^  we get  f  t h i s i s an i n t e g e r o f the 2nd degree i n  reduce to o r d i n a r y i n t e g e r s , and hence  j>  -x  i t must  ( G-, Cl^ ~  as the theorem Sub-case A;  a  o  f sTrurd. f  states. d> /Q- and $/S  e  By Theorem V, Cor. 1., i f there i s a maximal reduced, i n t e g e r of the 3 r d degree i n ^ but none o f lower degree, then there i s an i n t e g e r we f i n d t h a t  and  J-ff.  On  are s a t i s f i e d , w h i l e I,  r e q u i r e s the c o n d i t i o n that ft^ I £ if  ^  and  2nd degree i n  ^ V  s  substituting,  However, from Table I, #  there i s an i n t e g e r  -  of the  -*c which i s c o n t r a r y to h y p o t h e s i s .  Hence,  i n t h i s case, there can be no maximal reduced i n t e g e r o f 3 r d degree i n -x .  the  Sub-case B:  •£> j>& and j> 4>S.  In t h i s  case, by Theorem Y, Cor. 3 . „ i f there i s  a maximal reduced i n t e g e r of the 3 r d degree i n  but none  of lov/er degree, there i s a s i n g l e - p r i m e reduced i n t e g e r  where  <X ^ o 0  and  gruence of Theorem B,  CL =£ O . t  O.  A l s o , s i n c e from the second con-  a, a  x  = <a ,mod ^ 0  we must have  38  if  f^,  r,v  which i s impossible we  musjr have  since/ G  ana  ^  in  r e s u l t i n g congruence by  and  Q  we x  60,  Also We  (  may,  A l s o we  since  there f o r e  write  have  degree i n  == 0,  . Id,  ~  there .  zcz  0  =  How,  }  CL^Q. and  and  Q.  and  ^  than the f i r s t  ^—-  ^ *  Q."— 2  can  then from  o > ~ J  t h e r e f o r e , have  ±a, —  since  Q.*"- £S s 0_> mod  i s an i n t e g e r  i s prime to  Q. yO^(f>  higher pov/er of ^  We,  ^  .2 &o = a^Q.yrr^^  ^ for i f  x  Table I,  -c  Ho  Q -f*$  divide  - Q.S  / /6 S (a*- ^s)*"  jfe /(Gl*~-sj  D,  5  }  9  get  - a, a * i"-& - 3 Q s =  ^ I\  m u l t i p l y i n g the  3  j.cl,S  from  9  u s  ;  or, s i n c e  Hence  % as - a = 0,/yyi&vl f>  cx, o.  adding  are/b^oth odd*  /> 7^ i. •  Substituting  and from  S  o f the  hs  ,  2nd. where  ^-  substituting for  S-&  0  and  z A,  in  X  ?  we  get  - xa^(a -us) x  or, s i n c e  + &(a ~ L  a*"- US .= #•<.  tts) - X j> 4 and  * s) = 0,^yoL-f^  f> 4>-<-  %. CLi — d = o , sm<a?i jf)  0)  I^  S i m i l a r l y , from &x C ^ s >  we get -i- a a.  which., on s u b s t i t u t i o n o f  (g£-~us)  x  -1<x~^c(c£~~ h9)  Q. - ^5 =  reduces t o  whence, i n view of ( l ) ,  or s i n c e a  -i_aii  x  = O, ^^cLp  Making the same s u b s t i t u t i o n s i n  which, on w r i t i n g  whence if  %  Q.^- i+S = ^-c . ' becomes  J,  we get  - 40 -  Nov?, from ( 1 ) , la? =• Q., mod ^ S u b s t i t u t i n g these we  and t h e r e f o r e there 3 r d degree i n Sub-ease 0;  which  l  and cannot  $ 4> a • i s no i n t e g e r  ~r (Ci , d, , a.^ , f) 0  can be no maximal reduced i n t e g e r o f the  41$  but & ? Q-  I f there degree  O mod ^ ,  * f l s O,, mod ^  and  Hence t h e r e  a -k$=  since  s  a d d i n g these two we get $ ^ %•  mod |5  get  But t h i s i s i m p o s s i b l e  hold since  ^a^s  and  i s a maximal reduced i n t e g e r o f the 3rd  i n -x but none o f lower degree, by Theorem B,  i s a single-prime  there  reduced i n t e g e r  ^  =  ~p (&o > o., j a i i') x  such t h a t  a a" 0  x  =  - S  a, a = a z  From Theorem V , Cor. 2 . , C L i  -q  or  a,^ - o  cannot both be z e r o .  £ =0, 0  but from (3)» The  (0  V/rwcLf  0  a ? - * , * - *  j  (ZJ  J  ( ? )  and hence, from ( 2 ) e i t h e r since  jfr4>Q.  }  CL, and  only possible integers, therefore,  (cL) (hj  >h-tnd Ifj S =fc O Tn^d f ^ ' V -n^-o-d j S B  6.^.-/  ^  ^  -^^Y-y  ^  ^r-c^^-j  •  -TH-ereS If l<^j£c  }  g = •*-&,•<-/  S \c 0 rt-^rrl 2. •  ~  ^ ^ 2 . . ^ / ,  5 £ l>- y^r*{• If  - 41 -  V  are  JL<>,  The f i r s t  o, a. , / ) x  of these,  a  p  p  and.  f (o  }  s i n c e ft IS  }  however, cannot o c c u r .  we g e t 2>(i. s - Q / = C?, mod. ^  in  a, O, i) , a * ». For substituting  which^is impossible  A  Substituting i n and  ftf-CL  we get a , = a mod ^  Substituting this i n and  X,  , _T  condition. if  a  r  butft4>(X-  We need., t h e r e f o r e , only c o n s i d e r  ^/S  P ^-  , and  a  I 2"^  Hence there and If  (o,  a,/J  = Q^mcd ^>, or s i n c e ;  and t h e r e f o r e  we f i n d t h a t  3  ^ -  a, = a +• -kf-  £ = Oj mod ^ *  are a l l s a t i s f i e d by t h i s same , G-, j O,  i s an i n t e g e r  /)  o-,= Ql mod -ft>.  t h i s i s not maximal, there w i l l be a maximal  reduced i n t e g e r of the form  where  ^  >s3 and  sm  We c o n s i d e r  From Theorem I we have but  ^  ;  ,  ft***I/6S(Q. -fys) x  first  the case fi^fi±.  and s i n c e ^ / f i  X  ^ j>(Q' ~/ss) &ntiL hence f o r -p ^ z , i t f o l l o w s L  J  that ^ * 7 £ . A l s o from •we 'have  £, s  Z (P~) since S  Q._, mod ft*, and t h e r e f o r e  We have  or, since  ^/£  S  and  ^^/(a,-a)  and ^>^Q. — GL + p^Ji  - 42  -  a p a r t from o r d i n a r y i n t e g e r s . •p ^ -7c  where  integer  i s the g r e a t e r  ^  <o = -^.jfor  Therefore of  and  ^x.  i f not gives  'T^--* ^ > a c c o r d i n g as  o r  or ^ < -a,and as. n e i t h e r of these can e x i s t we  an  >  ^  must have <d = -t.  Hence  I  Substituting i n - a  X d  0  and  from I (f)  }  =  -3a Q+a a\ta S=  s  Since  ¥  0  t  £>, ^vo^L  p  and t h e r e f o r e , + aa x  a d d i n g t h i s to ( 4 ) , we  and t h e r e f o r e  get  a  0  -  ^)  Oj sryu*>(  x  ^  we  =  since  -A  get  ~ o  and  a  from (4)  = o.  %  Hence the  i n t e g e r reduces to  Substituting this, •J"  J.  and  1",  are  satisfied.  gives  5' -Q (£- -*/)  V- 5 ( W , -3Q.)  which i s s a t i s f i e d s i n c e £ Similarly  which i s a l s o Hence i f  f S  and  =  O sry«rd'£ )  a,- &~ 0 mod ^ ;  T  gives  satisfied. ft-f  ±,  we  have the maximal reduced i n t e g e r  -  (®  }  ®  y  such t h a t  43  -  where  J  ft "IS  and.  x  sn. i s the g r e a t e s t i n t e g e r  a, =  Q_ mod ft™  ft ^ £. •  We now c o n s i d e r ^ t h e case  If  {0 -^yOj^J  °^^<*i&  i s not maximal, there i s an i n t e g e r of the form  wher^  ^i^"^3  and  . From  -3Qa  o  I (£)  i f^  s  -f £* \^ <2 V *. S) £  •=  4  we have  Oysrhfret 1*''-  or  —-C6)  according aA  y^  or  divisible  Neither ( 3 ) nor\(6),ean  ^  hold unless  f o r i n each, one term is  by  w h i l e the other \s n o t .  have e i t h e r  ^ =A  or  a  0  = *  f c  -3Q cl a  and e i t h e r l a t t e r case  B  ^ o  0  $4?om J. (dt-) ^-jft s  %  and a^^. o  or  a  0  — a*_ = o .  %  ±s-^phe  i s c o n s i d e r e d l a t e r ^we—44rS*ega*d—i-t~he-3?-e-. 0  and  a ^o %  .!>  we get  t h i s i s i m p o s s i b l e unless  Hence the o n l y p o s s i b l e i n t e g e r i s  from which  ^^yJUrdJ  r  +C& +%&)&z=0,'^*^%.'  0  Substituting i n  a ^o  Hence we mdst  = o. \  F i r s t we c o n s i d e r the  Since  ^  }  —/  But  (_0 , x ) O j -%_y  we get an i n t e g e r (-  t h i s from ^ degree i n  i s a l s o an i n t e g e r , and s u b t r a c t i n g  ^  i- l=>  a(&,-Q.)  ¥  in  z ~~*'Q. '+ i  XJS  «  /£  + dS  £^l(a -G) t  * S  5=1  and i f  we get  3  (a,-a)  and  4.  i.  i  us  i  therefore, since  integers.  c,- Q. = 0, mod 1 -  and  Substituting  Hence  o f t h e 2nd  which must reduce to o r d i n a r y  Hence we have  and  s O>  =  = O, ^nurd  X .  we have  £E — Q.  Similarly  from  I*  since  ^  $0'  —  find  3 & +a  Z-  and  we  ,  X $ O.  and  <?, smovL  -£ ,  X  x  + Q$-'Q^s'^  // mod 4-^  o  -asd~*»©ffl-44)——^-W^^se-d-^,  we—get—  eaad-  J J L i s evidently s a t i s f i e d .  S u b s t i t u t i n g i n I,  get  1  (&-/)•  W r i t i n g ft, = Q + z  f  Z. l  S[X  •  • JL  Q(a,-Q)  and  (l + QSj -hi &A +x-£•]  +  5 = iQ.4L-+ X -hi  X  this  S(a,-Q)  becomes  A Q S as ==O muytiX y  we  -  45  -  In view of ( 7 ) , we may w r i t e / ffis' =  s' = */-Q  and s i n c e  and t h e r e f o r e  Q = /mod >^  we have  1  I + Q.9' = O, /rrurct HHence the f i r s t term of (8) i s d i v i s i b l e "by  and may  be omitted, and the congruence becomes i  1  A Q. + 1  4Q £ •/- Z. 5 {a,4 + L s' j + X 5'=  We now s u b s t i t u t e  • -2.  or s i n c e  M-€ - Q. ,  QS = 0,/moxLl  and on d i v i d i n g by  we get  a., -J^^uM^f,  f-i  x  d) =0 ^v<rc(^ }  which i s e v i d e n t l y s a t i s f i e d . Hence i f S  X. 5  A  where  i s a maximal reduced i n t e g e r ^  S s  mod  > - j - , ~ > j-^J  We now c o n s i d e r the case  £<, = <3. = o x  there where  If)there  i s an. I n t e g e r  ^  then  .  /  1  ar  i s of the 2nd degree i n integers.  .  r  and hence must reduce t o o r d i n a r y  Therefore we have  Substituting i n  1  ,  w  3  e  #••<,.,..•:.:  ± JS  and  a, =  Q, mod £.,  have  Q. (&,~ a) + us (a,-a) + as ~ o,srrt^ il*' x  *<-~^  and it  1~JS  therefore,, since follows  the  and  that  S i m i l a r l y , w e f i n d that while  a, = e.moa  and  Jj_ and  i n t e g e r (o  }  2%  1,  i s s a t i s f i e d by the same c o n d i t i o n  are i d e n t i c a l l y z e r o .  ~^ > 0  i n t e g e r such t h a t  }  j ^ } where  iV""/£^ and  --n  i s the g r e a t e s t  a, = & modi.'"-  o n l y be maximal when the i n t e g e r f£ > -ri •> e x i s t , ^ since i n the l a t t e r SUCh that  /JJy^. ^^Cu^Cl^^ Sub-case D;  ^  j/Q b u t ^ ? 5  > -p,J  l^-e -t^i^^  CA^L,  1 ) ^Pfrj^-e. ryu^i^aj?  a  This  i s the g r e a t e s t  A-i^o , i-^^iJLla.  -  Hence we have  /  will  f a i l s to integer  AJZ^-Z_  (2-^/  <z^l~c*f»^4 ^^C^iZ^  Lst_  s  .  From Theorem B, i f there i s a maximal reduced i n t e g e r o f the 3 r d degree i n  but none o f lower degree,  there i s a s i n g l e prime reduced i n t e g e r /  = f (  a  ° >> 1 ± a  a  >  0  such t h a t &o <X = - S X  snwrfft>  =  do ^nvtrd. -ft -  =  o  t  since  fa  &  x  - a,  ^y^ol^  Q)  (Z) and  0)  From Theorem 7 , C o r . 3., ( or from congruence ( l ) , above), we have (2)  a„^o  and  G. ^t 0  or ( 3 ) , we have a l s o  x  }  and, t h e r e f o r e , from e i t h e r  &, =£- O  From Theorem I , we have therefore  since ft I OL  but  and ,  we must have jf)= z •  - 47 -  Hence  a  0  — &, = a.< = /  and the i n t e g e r  x  7 Substituting  ^  satisfied,  (i • i  J  *  ' i)  i  i n I,  Z  }  H  while  becomes  we f i n d t h a t  I- i s f  gives  z -f-Q + 4 5 -fas ~=- 0, ^rt^trt/ /f OX!*  (%. -f- a) (i + s)  which i s s a t i s f i e d .  From  or, s i n c e  and  }  ^vwcrfC h ,  2 , a f t e r c a n c e l l i n g and o m i t t i n g  terms d i v i s i b l e by 8, we get also satisfied,.  = o  4  **(S*+r) s  Substituting i n  I,  X IQ.  / *S fd S  However, from Table I , t h i s maximal reduced i n t e g e r  j * f rt  X  but jfi $S  £  £>, s**uro( tf  IB. Ct-Zyn^fffff-  i s the c o n d i t i o n f o r a  of the 2nd degree i n ^ y ^ x ^ w  i s t h e r e f o r e c o n t r a r y to the hypothesis  Hence i f  •  5  therefore  there  i n t e g e r o f the 3rd degree i f there  o f Case I I I .  ^/-^  can be no maximal reduced i s none o f the 2nd degree.  The r e s u l t s o f Case I I I are summarized i n Table I I .  which i s  we get  whence, since 1 M ,  and  0,mod S"  TABLE I I . Maximal Reduced. : Integer  Conditions  Sub-case A:  it>lQ.  There can he no maximal reduced i n t e g e r of the 3 r d degree i n x i f there i s none of the 2 n d degree.  and//£.  Sub-case B:  There can be no maximal reduced i n t e g e r of the 3 r d degree i n -x. i f there i s none of the 2nd degree.  & £ a and £ j>£ Sub-case  . ••  G:  that  2  ^ ^/s, and  such  a, = 0 , m o d ^ ^  Z  r  »  •. 1  /yt i s the g r e s t e s t i n t e g e r  f^Xj  I Is but i j>Q r  f o r Existence.  1  x  •  S — i.  3  mod t+, I.  (2)  ^  , -<n = X, where  and  =  5 S -a .  Q_ mod X .  aA0 &-5/ }  unsatisfied.  ^  the g r e a t e s t  i n t e g e r such that l ^ / s and Sub-case B: but ^ 5 " .  modi."!  • j  There can "be no maximal reduced i n t e g e r of the 3 r d degree i n of the 2nd degree.  i f there  i s none  

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