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The determination of sets of integral elements for certain rational division algebras Hull, Ralph 1930

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U . B . C . CAT tn '] ACC. MO THE DETERMINATION OF SETS OF INTEGRAL ELEMENTS FOR CERTAIN RATIONAL DIVISION ALGEBRAS.  by  Ralph Hull  A Thesis submitted for the Degree of MASTER OF ARTS in the Department of MATHEMATICS  The University of British Columbia March - 1930.  A  LtBRARY  Af^i. /?3°/?s ^ ^ ^ /L  TABLE OF CONTENTS  1.  Introduction.  2.  The General Dickson Algebra of Order 9.  5.  The Rank Equation for the Algebras D.  4.  Necessary Conditions for Integral Elements of the Algebras D,-Congruences.  5.  A Classification of the Algebras D.  6.  The Congruence  7.  Reduction of Congruences.  8.  Complete Solution of Congruences,  9.  Sets of Integral Elements for Special Cases.  ^ - ^ ^ y + Z -ofrrroofd)  10.  Sets of Integral Elements for the General Case.  11.  Appendix: I.- Tables for Computation. II.- Substitutions in a Cubic Congruence.  12.  Bibliography.  1  THE DETERMINATION OF SETS OF INTEGRAL ELEMENTS FOR CERTAIN RATIONAL DIVISION ALGEBRAS. 1. Introduction. The purpose of this paper is to determine integral elements of a certain associative division algebra^, D, of order nine over the field of rational numbers. basal units of D are  with d  j = 0 <2)  The nine  where  a rational integer having a rational prime factor of  the form 7fT7±2  or *?f77:t ^  power which is a multiple of 3.  that does not occur to a Also, X  satisfies the cubic  which is cyclic, i. e. it is irreducible and has roots ^ (f  ,^ ,  , of the forrn^  Further, A-y = where /If'  *  It follows that  satisfy f/J ,  and that ^This is a special case of the algebra of order over a general field F, discovered by Dickson, and called by Wedderburn a Dickson Algebra. See Dickson's Algebras and Their Arithmetics, p. 66. 2 F. S. Nowlan- Bulletin of the American Mathematical ,Society, Vol. 32, p. 575 (1926).  2  Integers of the cubic number field, ^u^^ , defined by a root of  are of the form  ^ with  Z = ,  + <*, X f  , and ac^ rational integers.  Professor F. S.  Nowlan"*" has shown that the norm of a prime of  not  associated with a rational prime is either J , or a rational prime of the form 7/77J!!:/ .  In case the prime is associated  with a rational prime, the norm is the cube of the rational prime and is thus of the form <?/77±/ . than J  , or those of the form  Rational primes other  are primes of  Further, every rational prime 7/77^/ is factorable into three conjugate primes of of  , and so is the norm of a prime  . Thus the restriction on ^  insures that D is a division  algebra^. In future developments, we shall need a corollary to the above, viz: fj)  The form, F, given by f =  ^ 6 -  -3  f J or, <3^ -r  or/  of the norm / V ^ o f the general integer,  ,of A'^represents  / , J , all primes of the form 7/77^ /,and all products whose prime factors are J? and primes  .  The form represents  ^F. S. Nowlan- Bulletin of the American Mathematical Society, Vol. 32, p. 379 (1926). ^Algebras and Their Arithmetics, p. 68.  5  no prime other than these.  Other primes are divisors of the  form only when they divide each of  , and  and then  they appear in powers which are multiples of J . The general element Z of D satisfies a rank equation*** viz:  a certain cubic equation having unity for the coefficient  of the term of third degree and having its other coefficients rational integral functions of the coordinates of z .  More-  over, this element does not satisfy a like equation of lower degree. The integral elements of an algebra are defined^ as those elements belonging to some one set possessing the four following properties: R (rank):  For every element of the set, the coefficients  in the rank equation are rational integers. C (closure):  The set is closed under addition, subtrac-  tion, and multiplication. U (unity): 1.1 (maximal):  The set contains the modulus 1. The set is maximal, i.e. it is not contained  in a larger set having properties R, C, U. We restrict the investigation to those sets which contain the nine basal units  y'*^ .  We write ^  where  only prime factors J or those of the form ?/*r7 2r/ has only prime factors of the form  or  contains  , while 6 ^ at least  one of which occurs to a power which is not a multiple of 3 ^Algebras and Their Arithmetics, p. 111. ^Algebras and Their Arithmetics, p. 141.  ;  and, further, 6 itself is not of the form Sri?^/ .  A trans-  formation of units introduces the new basal units with  —^  .  Then for each of the cases,€-?/T7t2,6 = 9/77i:^,  there is a unique set of integral elements which contains the original basal units  2. The General Dickson Algebra of Order We shall consider the associative algebra G of order 9, defined as follows. y  Its nine basal units are given by  *  integer.  ' where y^- J , with ^  Further, X  a rational  satisfies the cyclic equation  whose coefficients are rational integers, and whose roots , ^  and  <f -  , and  <!°  , are such that  , where cy , <b , and c are rational integers.  Also A y - yx'  w^ere  It is readily seen that and  x"  satisfy the given cyclic equation, and that A The associative law and ^ x'y - * A y  imply -  ^ y*'^  ^The material of this section is in part given by F. S. Nowlan, Transactions of the Royal Society of Canada, Third Series, Vol. XXI, section III (1927), pp.187 to 188., and is here incorporated for future reference.  5 and in general  Now let <xf be any polynomial in x denote the result of replacing respectively.  X  , and let  and of"  in <=< by x' and A''  Then a like procedure gives ^ y = y<*r' ,  ^  y^'^  The general element of the algebra G is Z = /!  ,  where  s  = or.jr  ,  =  ,  (I = the coordinates ^  ^ ^ X 1- r, * + r,  , ^^ , and f^^tL^Oj', zjbeing rational  numbers. The rank equation for algebra G^ is  In terms of the coordinates of<Z , the coefficients of and  , where  is the discriminant^ o f ^ ,  this becomes^  ^F. S. Nowlan- Transactions of the Royal Society of Canada Third Series, Vo&. XXI, Section III (1927), p. 189. 2 The necessary and sufficient condition that the equation be cyclic is that its discriminant be a perfect square. Loc. cit. p. 186. ^Loc. cit. p. 192.  - f Joe. -  +  7  3. The Rank Equation for the Algebras D. From the rank equation for the algebra G as given in the last section, we obtain that for the algebras D (defined in the introduction) by substituting O , - 3 , / , for f ,<? ,and /? respectively, and noting that the discriminant of f/ =<7^ so that for D,  ^  - 7  /^ ^ 6 ^  -J  -3  We get  ^ \ j^  -J  jr T7  ^  .  <H J ]  is  ^  ^ ^ ^ ^  ^  ^  ^ ^ 7 V, ^ ^  =3,  -3  ^  8  4. Necessary Conditions for Integral Elements of the Algebras D, - Congruences. To determine integral elements, defined in the introduction, we first stipulate that the coefficient of ao^ in the rank equation,^F) , shall be a rational integer.  Thus we  require  with  rational integer. From closure,ZA andZA'are integral if Z  is such.  To  obtain the rank equation f o r Z A from that o f Z , we have 2r/r -  ^ yRx  =  /- y ^ C x  . y B, ^ y ' C ,  where, from ///  and, similarly, c, ^  j ^JLx ^ ^  Thus we obtain the rank equation for Z A from that of Z  by  replacing:  ^  -'  (3, " (3. ,  ^  "  ^  ,  1  "  ^  ^  -  r,  ^  From the coefficient of ^ ^ i n the rank equation for Z/f we require  9  with // , a rational integer. Similarly, for the elementZA ^ we require  Likewise, the elements if Z is integral.  are integral  By a method similar to that for Z X  simplifying by means of  and remembering that  , ^ , we  obtain the six additional relations  =  <rj -  rj  - 3 ^ f / ?<rj  =  where the  K  ^ / ' s a n d W ^ a r e rational integers.  On solution of e q u a t i o n s ^  a n d ^ - ^ ,  we obtain ^ ^  ( y^, ^  //^ ^  ^  -^ ^  ^  ^  (<7f(3,  = //K  ^ -<, K fzo)  7  ^ // M/.  ^ ,  - z t/ - <, L/ ^  f  ^  11 K  ,  -vH/,  10 We thus have Theorem I.  A necessary and sufficient condition for the  coefficient of  in the rank equation for the algebras D to  be a rational integer is that each of 7<?/o , ?<=/, , 7  . 7 ^ ,  , 7<f<r, , and  We employ of ^  ,  ,  ,  be a rational integer.  , and^<?y to express the coefficient  , and the constant term, i.e. /Vfz), in the rank  equation of Z  , in terms of the /7s, ^ s ,  andt/V^-s.  By means  of tables listed in the Appendix, we find the coefficient of to be as follows  Likewise, /Y^jbecomes  ^  tf' K -  ^ y  -  sttt/Lf^ vjMKJ/K'  ^ ^  ^z/  ^  ^  11 Since these are to be rational integers/? 21 is to have property R, we have Theorem II.  A necessary condition that an element <Z. of the  algebras D be integral is that congruences  and  which follow, be simultaneously satisfied.  - (-Km  V I K %  ^ 3 tf'K - K ' J  ^ ^ K, H i -i  f  ^  ^  12  5. A Classification of the Algebras D. We write G, where ^  has no rational prime factors other than ^  those of the form 7/77 tr/, while the prime factors of of the forms <7^7  2 and 7/77  , or ^  are  with at least one of its  prime factors occurring to a power which is not a multiple of ^  . Seven cases arise: Case 1.  - e  with  Case 11.  6 6  Case III.  - 6  "  6  Case IV.  r - 6  "  ^  Case V.  y  " n  6  Case VI.  -  &  Case VII.  with no restriction on  that stated in defining it. In this paper, cases 11$ - VI are fully considered, and case VII is considered with 6 Case VII with €  restricted as in cases III - VI,  so restricted is, in this paper, hereafter  referred to as the general case.  6. The Congruence We shall have frequent occasion to use the results here obtained from the congruence  13  where (f ^ ^r/ From  From which ^ r - ^  y-z  -  6  .  Similarly z -A  - ^ J/ - Z  Whence  Also, squaring  ^ ^  x^^yz ^  ( y z t z ^ ^ f^y-zj Y ^ p o ^ /  In like manner, we obtain, using ^ ^  since  x*- ^ y z =  y- z x  -  ^ ///77<3c/jj.  7. Reduction of Congruences. ^  We proceed to the reduction of congruences for cases I - VI with <f- 6 with  .  From ^  , and^-^  , (22J , and  we have  = 72  f - (3,  ^f  ^ ^  ^  ^  <  14  Since from  , for these cases we require  we must have, from the corollary to the theorem stated in the introduction // ^  ^ - </ K  = 3  ^  f ^^  ^  t/  =<7  - ^ %  ^ ^  ^  t/  =0  since the rational prime factors of  7  /  6  are primes of  We have / /  whence  - T-  -z  z  -y  /  ^ s ^ =  -  ^  / 2 f^lOc/ejL  Likewise we may show, for cases I - VI with =  VI/ =  <pjr 6  ^ o c / e /  For cases I - VI we accordingly write  ^  )  - ft.,  VL  ,  .  that  15  On making these substitutions, congruences  and  become for cases I -VI,  - J 6 {-<-/. nty. iLj/y.  ^ ^  The following combinations of the nine unknowns prove convenient.  We let  r - z/^ ^  j - r^r'  ^  ^  s - ^ ^ ^  ^ -  ,  ,  In terms of these new tariables /j?/^  f - t ^ ^ n A ,  -  7* f t '  becomes  16 Likewise.  becomes  f 6 ^ r '-3 r/ r - J // ^ / z ^ r - f  -  /  f i t tA^.  -J  t^ - 2T ^ ^ - ^  By inspection  while  p^)  since  ^cr t/v^ ^ ^  - ^^  H/J  requires  requires  , Applying the relations of section 6 to  Whence from  That is  we have  we have  17  In view of  fJ?<?/  f  we write  -Jt,  ,  /f/'  7/77 ^ r ,  where f77— o^ Substituting  ^ ^  in  , cancelling and  grouping, we require  73/77 Y whence, since  = /  condition that  (/J^*  , we have as a necessary  be satisfied  Then, from  Equations  ^ ^  and  , according to the relations  obtained in section 6, yield  We now substitute  in  ^ ^  .  Aft er  cancelling and grouping***, we obtain as a necessary condition ^The result of the substitution is given in full in the Appendix.  18  that  p* ^  be satisfied -J  8. Complete Solution of Congruences,  ^r/^oa^^.  We consider first cases III and IV with For these cases  Hence for cases III and IV, using  ,  (^fJ?) and  ^ ^  'and writing congruence  Let  takes the form  rr7 — O  If, in we have  *  ,  Then  ]K=0  ^  requires  , then  ^oc/j).  =0/^700(3), a contradiction.  If  t/  19  Hence if 17 - O ^ Let  rr7 — /  .  ^ If  X  Then  /  ^  = requires  V-^rr-A  , we have  /EE <3  If A ^ ^ / , we have has no solutions. Let  /i7 — - / . -  If  p  -A'^of'T? , a contradiction.  - !K-/ EE  Hence  m  Then  ^  ^  /? ^  f/77 0 3* J,  v.hich  /  requires  vA ^ -x  , we have  Hence  - / = <P  BO / ^ c J ? ; .  , a contradiction.  Hence  o If A ^ ^ / , we have has no solutions.  which  Hence  r?7 ^  —/  Thus for cases III and IV, we must have X That is from (r/)  =  r - o  /77 -=<2  f/770</^.  , with /77—a,we require r f r ' =5  S'J =  ??' ^ c / m o ^ j ? ) .  From ^  ^ - r ^ o c / J-J.  From ^ Combining ^ ^  ^ and ^ ^ r = e 3  s o ff77 0 3/jJ.  , we have = f ^liy^JjJ.  For cases V and VI with df ^ ^ y//77<96/<?^we have  =  - 3  and  20  By an argument similar to that above, we may show that for these cases also, the congruences  ,  , and  hold. That is, for cases III - VI we have as a necessary condition that  be satisfied r =  From  ^  and  ^  EE  we have  For cases III - VI, from  ,  and  we may  write ^ ^  r - j / ; 7-/77  t = J/; ^ where  /17 —  ,  ^  r'-  J r-  or -/.  le substitute ^ ^  in ^ ^  //77(9<//f/)  and obtain  on simplifying and grouping  ^  21  /-/??  7/77 ^ ^ ?  M/n)  - 3/77 Y '  As is shown in section 6,  ^  ^  yields  We make the following grouping of the variables  whence  ^  reduces to  - ^  In  ^  , which follows;  -  7/r  ^  f ? ^ ^r" - ^ / r r ^ p ^ ^ r r  For cases III and IV with  rr?  .  Then  has only the solutions Let  ^  we write  and thus require ^  Let  ^ ^77  rr/- /  .  Then  %  <f =  ^jy!  - XjK^- V = o /r^ c c / j ^  gives  which  K^o/zTy^^/j?/' ^ ^ ^ = /f'^Fd'j^  which  has no solutions. Let  /77--/ .  Then  =  which  22 has no solutions. Hence for cases III and IV, we must have X  E o  and  .  That is, from ^  /77  and  , ^ ^  ,  ,  , we have rr-3 ^  ^  3  J 3, ^  /-=  and  A similar argument shows that cases V and VI with  ^ ^  6  -  hold for  .  By inspection the quadratic congruence, to be satisfied identically by ^ ^  and  ^  , is seen ^  From the results obtained in this section and section 7, preceding, we have the following: Theorem III. element  Z  the forms <7  The necessary and sufficient conditions that the of one of the algebras D with -t" <2-  that relations  ^ ^  or <7/77^ y -  ^ - ^  , and  of  , shall have property R, is  above shall hold simultaneously.  9. Sets of Integral Elements for Special Cases. For convenience we make the following grouping in considering integral elements for cases III - VI.  7; =  6  r 7/.  Let  23  Then  requires ^  / i ^ e S , t 7J  In terms of r .of  ^ ^  , 5 , t , r', s', and  Z  the coordinates y r ^ ^ y  - /t^  21 possesses property R, are given by  -  yr' 7 r  7 (3, ^ 7 (6? ^ 7 ^  z S'f25  -3 K ,  y s ^  -3 ^ ,  ^  ^ -  and  — 3  /r'  ^=  .  ^ ^  these reduce to  - ^ ^  — JL^L -  ( °  r ,  -3 ^  ,  ^  ^  -z 5' - 2 s ^  7/2 r  In view of  -  f  ^ -. z 5, 3  - 3 -  3 3  3  S, J  s/.  ^  3  ^  3  z 7; J 3 3  24  Further  Z  is given by  where  has integral coordinates, and  As shown in the last section, the following relations must hold  7 ^ S, ^ 7;= 3  "  .  For cases III and VI, where 6-7/777*2. and <?/?7-7 respectively, we have  (9  -/  and  becomes for these  cases  - 5, t v ; A typical value of  obtained from  jy).  , for these cases is  by placing  *  ^  — O^and  /p rr ^  —  /  j  - these values being such that relations  are satisfied. From  y Multiplying  on the left by X  =  - /-  and on the right by  we obtain  and  respectively, whence  Similarly  Further  -  T/  26  Similarly  Also, using relations just obtained, we find that f ^  y ' rr - y -  y >f  By means of relations given by  , in terms of  Referring to  ^  -  ^ ^  .  Y<e get  it is evident that  expressed linearly in terms of rational  ^  ^ ^  we express  -Z/ ,  <2, , can be  ft^j — o^,/, ^  with  integral coefficients.  The rank equation for  , is 3  _  o  That is  where  3/77- /  since ^r =  The results just obtained show that we have, for each of cases III and VI, that is for the cases ^ - f / ^ y z respectively, a unique set of elements possessing the rank property; R, and having the basis  , which  contains the original basal units y ' ^/^'/J  .  existence of the basis shows that the set is closed under addition, subtraction, and multiplication, and the set is  The  27  obviously maximal since it contains every element whose coordinates satisfy  ^  do not satisfy  , and no element whose coordinates  belongs to the set.  Further, the set  contains the modulus / .  Hence for  or <?/77 — y  (i.e. for  ), we have a unique set of integral  ,  elements according to the definition given in the introduction. For cases IV and V, that is for (f-=%/77-2 or ?/77 we have  (f ^  / ff77<9c/ J^  ^  .  ^  For these cases  % =  ^  ^/77<7</jJ  becomes ^  V S , V-7J  A value of  , is  Proceeding as for cases III and VI, we find ^ ^  y  - / ^  ^  x  -  K  y' ?  x'H  ^  K  r'^  By means of relations cases IV and V, as follows:  , ^,  -/  ,  , we may express Z ^  , for  ,  28  y Evidently, in view of in terms of  t  i^^-jZ!, , is expressible linearly  ^ ^ ^ ^  with rational  integral coefficients. The rank equation for  ^  is  That is ^ ^ where  — <0 ^  /T7, is a rational integer since e thus have for each of cases IV and V with  and 6- — r e s p e c t i v e l y ,  a unique set of integral  elements for which  ^^  ^^  forms a basis.  10. Sets of Integral Elements for the General Case. As stated in the introduction we investigate the general case, VII, with  (f restricted as in cases III - VI, i.e.  or 6 — 7/77 e first consider the effect of the introduction of in place of the basal unit  y  ^  is a number of  y  — (f, ^ <f, ^  ^  ^  .  of an algebra D, where  Then x  =  y ^  '  .  Let  29  as in section 2. Thus  v ' ^ and the substitution has the effect of dividing For case VII, factors 3  6  with f>  , and those of the form  the norm of an integer of Let , ^  /j7 - / y ^ ^ !  where  by^^-  containing only prime  <?/77^r /  ^ ^  <T  .  Thus  ^  ^f - e.  and  , and ^^ , are rational integers.  We now let  y - y,  .  By the result obtained above 6  The general element  /  Z*  is  - a  becomes  'f ^ f t ' T ^ ^  * ^  or  where  ^By the corollary to the theorem stated in the introduction.  A')  30  ^^  For (f — f/77J!^2  or  ^^  <?/77_fr y  , we have the conditions  of Theorem III of Section 8, but with reference to the basal units  J^'  /, z /  We see that y - x  ^  ^^ * ^ ^ * ^  is a linear combination of the new basal units  ^^  with  rational integral coefficients, and likewise for the other basal units  ^^  .  Hence sets of elements satisfying the  conditions of Theorem III will contain the original basal units  y'  .  The results obtained are summarized in Theorem IV.  For the algebras D, with  (f  where  fl  is  31  the product of positive integral powers of J? and like powers of rational primes of the form 7/77 2!:/ , and where product of rational primes of the forms ^<77 Z? Z  6?  is a  and 7/77 ^r^/ ,  with at least one of these primes occurring to a power not a multiple of  -3  , with the further restriction that  is of the forms 7/77 TL 2 of  (f  or ?/77  y  itself  , we have for each value  satisfying the above conditions, a unique set of  integral elements.  In each case considered, the set contains  the original basal units  52  11. Appendix. I. Tables for Computation. We give here a partial list of the expressions for powers and cross-products of the or'^., the  , t/^  , and  ^  f/  ^  ^ ^  , in terms of  .  = f/y/ -  , and  A; ^  '-  z^z/ - ^  ^ ^ ^  ^ ^  7  7  // ^ /z  y Z/^  HH', -  ^ Z / ^  ^  ^ ? JZ/ '  Z/^  -J  /yi/a;  ^ t/14  K H^  34  II. Substitutions in a Cubic Congruence. The following is the congruence obtained on substituting in  .  The simplified form is  given on page 17, section 7.  f 7/^-6^777^5 - 6 ^ / 7 7 ^ - 3 ^  777 ^ ^ 6 / 7 7 ^ ^ -  K ^ j  7/77'/; / m " - 36 ^ ^/77 - J  J 6  6/77 V  -^J/77^,  ^ 6 777 ^  777777)-5 777777^  y-^777  -  777  / / J/77S  *  ^  35  12. Bibliography.  1.  L. E. Dickson - Algebras and Their Arithmetics. The University of Chicago Iress, Chicago, Illinois.  2.  F. S. Nowlan  1923.  - Bulletin of the American Mathematical Society, Vol. XXXII, No. 4, July August, 1926.  3.  F. S. Nowlan  - Transactions of the Royal Society of Canada, Third Series, Vol. XXI, Section III, 1^27.  

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