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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 . L t B R A R Y CAT tn Af^i. /?3°/?s ^ ^ ^ '] ACC. MO /L 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 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 ^-^^y + Z -ofrrroofd) 7. Reduction of Congruences. 8. Complete Solution of Congruences, 9. Sets of Integral Elements for Special Cases. 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. The nine basal units of D are j = 0 <2) where with d a rational integer having a rational prime factor of the form 7fT7±2 or *?f77:t ^  that does not occur to a power which is a multiple of 3. 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 ^ Z = + <*, X f with , , 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±/ . Rational primes other than J , or those of the form are primes of Further, every rational prime 7/77^/ is factorable into three conjugate primes of , and so is the norm of a prime of . Thus the restriction on ^ insures that D is a division algebra^. In future developments, we shall need a corollary to the above, viz: The form, F, given by fj) f = ^ 6 - 3 f J or, <3^  -r - or/ of the norm /V^of 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 follow-ing 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): The set contains the modulus 1. 1.1 (maximal): 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 contains only prime factors J or those of the form ?/*r7 2r/ , while 6 has only prime factors of the form or ^ 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. Its nine basal units are given by y * ' where y^- J , with ^ a rational integer. Further, X satisfies the cyclic equation whose coefficients are rational integers, and whose roots , ^ , and <!° , are such that and <f - , 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 ^ imply x'y - * A y - ^ 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 , and let and of" denote the result of replacing X in <=< by x' and A'' respectively. Then a like procedure gives ^ y = y<*r' , ^ y ^ ' ^ The general element of the algebra G is Z = /! , where = o r . j r , s = ^ ^ , (I = X 1- r, * + r, the coordinates ^ , ^ ^ , 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 is f/ =<7^ so that for D, - 7 . We get ^ / ^  ^ 6 ^  - 3 ^ \ j ^  ^ ^ ^ ^ ^ -J - J ^ jr T7 ^ ^ ^ 7 V, ^ ^ -3 ^ ^ <H J ] = 3 , 8 4. Necessary Conditions for Integral Elements of the Algebras D, - Congruences. To determine integral elements, defined in the intro-duction, 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 forZA from that o f Z , we have 2r/r - ^ y R x /- 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 are integral if Z is integral. By a method similar to that for Z X , simplifying by means of and remembering that ^ , we obtain the six additional relations <rj = - r j -3 ^ f / ?<rj = K where the ^/'sandW^are rational integers. On solution of equations^ a n d ^ - ^ , we obtain ^ ^ ( y ^ , ^ / / ^ ^ ^ - ^  ^ ^ ^ , ^ (<7f(3, = / / K - z t/ - <, L/ ^  ^ -<, K f ^ 1 1 K , fzo) 7 ^ // M/. - v H / , 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 be a rational integer. We employ , , and^<?y to express the coefficient of ^ , 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. - ( - K m 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 ^ , or those of the form 7/77 tr/, while the prime factors of ^ are of the forms <7^7 2 and 7/77 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 6 Case 11. 6 Case III. - 6 " 6 Case IV. r - 6 " ^ Case V. y " 6 Case VI. - & n 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 restricted as in cases III - VI, Case VII with € 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 ^ ^ x*- ^  y z = y - z x -since ^ ///77<3c/jj. 7. Reduction of Congruences. ^ < We proceed to the reduction of congruences , and^-^ for cases I - VI with <f- 6 . From ^ , (22J , and with we have = 72 f ^ f ^ ^ ^ ^ - (3, 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 7 / since the rational prime factors of 6 are primes of We have / / - z - y - T-z / - ^ / 2 whence ^ s ^ = f^lOc/ejL Likewise we may show, for cases I - VI with <pjr 6 = VI/ = ^ 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, -J6 {-<-/. nty. iLj/y. ^ ^ The following combinations of the nine unknowns prove convenient. We let r - z/^ ^  ^ s - ^ ^ ^  , f - t ^ ^ n A , j - r ^ r ' ^ ^ - , - 7* f t ' In terms of these new tariables /j?/^  becomes 16 Likewise. becomes f 6 ^  r '-3 r/ r - J // ^ / z ^ r - f - / fit tA^ . -J t^  ^cr t/v^  - 2T ^  ^ - ^  ^ ^ - ^ ^ H/J By inspection requires while p ^ ) requires since , Applying the relations of section 6 to we have Whence from we have That is 17 In view of fJ?<?/ we write f -Jt, , /f/' 7/77 ^  r, where f77— o^ Substituting ^ ^ in , cancelling and grouping, we require 73/77 Y whence, since = / (/J^ * , we have as a necessary condition that 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 takes the form Let rr7 — O * Then ^ requires If, in , ] K = 0 , then ^ o c / j ) . If t / we have =0/^700(3), a contradiction. 19 Hence if 17 - O ^ X = Let rr7 — / . Then requires ^ / V - ^ r r - A -A'^of'T? If ^ , we have /EE <3 , a contradiction. Hence If A ^ ^ / , we have - !K-/ EE f/77 0 3* J, v.hich has no solutions. Hence m ^ / Let /i7 — - / . Then ^ requires - / ? ^ v A ^ - x BO / ^ c J ? ; . If p , we have - / = <P , a contradiction. Hence o If A ^ ^ / , we have which has no solutions. Hence r?7 ^  — / Thus for cases III and IV, we must have /77 -=<2 and X = r - o f/770</^. That is from , with /77—a,we require (r/) rfr' =5 S'J = ??' ^c/mo^j?). From ^ ^ - r ^ o c / J-J. From ^ ^ so ff77 0 3/jJ. Combining and ^ ^ , we have ^ ^ r = e 3 = f ^liy^JjJ. For cases V and VI with df ^  ^ y//77<96/<?^we have = - 3 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 = EE From ^ and ^ we have For cases III - VI, from , and we may write ^ ^ r - j / ; 7-/77 , r ' -t = J/; ^ ^ J r -where /17 — 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 , which follows; - ^  ^ - ^ 7^7 ^ ^ In we write and thus require ^ 7/r f ? ^ ^r" - ^ / r r ^ p ^ ^ r r For cases III and IV with <f = ^jy! gives Let rr? . Then % - XjK^- V = o /r^ cc/j^ which has only the solutions K^o/zTy^^/j?/' Let rr/- / . Then ^ ^ ^ = /f'^Fd'j^ which has no solutions. Let /77--/ . Then = which 22 has no solutions. Hence for cases III and IV, we must have /77 and X E o . That is, from ^ , ^ ^  , , and , we have rr-3 ^ ^ 3 J 3, ^  /-= and A similar argument shows that ^ ^ - hold for cases V and VI with 6 . By inspection the quadratic congruence, , is seen to be satisfied identically by ^ ^ and ^ ^ From the results obtained in this section and section 7, preceding, we have the following: Theorem III. The necessary and sufficient conditions that the element Z of one of the algebras D with ^ - ^ , and 6 of the forms <7 -t" <2- or <7/77^ y , shall have property R, is that relations ^ ^ - 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. Let 7; = r 7/. 23 Then requires ^ /i^ e S , t 7J ^ ^ In terms of r , 5 , t , r', s', and the coordinates .of Z y r ^ ^ y - /t^ 21 possesses property R, are given by - yr' 7 r - ^ 7 (3, ^ -z 5' - 2 s ^ 7 (6? - z S ' f 2 5 - 3 K , ^ y s ^ ^ - 3 ^ , 7 ^ ^ - r , 7/2 r -3 ^  . In view of , and ^ ^ these reduce to ^ — - ^ ^ — JL^L -3 - 3 - ^ /r' f 3 3 ^ = ^  - . S, ( ° 3 z 5, 3 J s / . 3 ^ - z 7; J 3 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 respective-ly, we have (9 -/ and becomes for these cases - 5, tv; jy). A typical value of , for these cases is obtained from by placing * ^ — O^and /p rr ^ — / - these values being such that relations j 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 - ^ ^ we express -Z/ , given by , in terms of ^ . Y<e get Referring to it is evident that <2, , can be expressed linearly in terms of ^ ^ ft^j — o^ ,/, ^ with rational integral coefficients. The rank equation for , is That is _ o 3 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/^yz respectively, a unique set of elements possessing the rank property; R, and having the basis , which contains the original basal units y ' ^ /^'/J . The existence of the basis shows that the set is closed under addition, subtraction, and multiplication, and the set is 27 obviously maximal since it contains every element whose coordinates satisfy ^ , and no element whose coordinates do not satisfy 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 becomes ^ ^ % = ^ ^/77<7</jJ ^ 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 , we may express Z^ , for cases IV and V, as follows: 28 y t Evidently, in view of i^^-jZ!, , is expressible linearly in terms of ^ ^ ^ ^ with rational integral coefficients. The rank equation for ^ is That is ^ ^ - — <0 ^ where /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 of an algebra D, where ^ — (f, ^  <f, is a number of . Let y ^ ^ ^ . Then x = y ^ ' 29 as in section 2. Thus v ' ^ and the substitution has the effect of dividing <T b y ^ ^ -For case VII, 6 with f> containing only prime factors 3 , and those of the form <?/77^ r / . Thus is the norm of an integer of ^ ^ ^ Let /j7 - /y^^! where ^f - e. and , ^ , and ^^ , are rational integers. We now let y - y, . By the result obtained above 6 - a The general element Z* becomes / 'f ^ f t ' T ^ ^ * ^ A') or where ^By the corollary to the theorem stated in the introduction. 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 6? is a product of rational primes of the forms <^77 Z? Z 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 itself is of the forms 7/77 TL 2 or ?/77 y , we have for each value of (f 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'^ ., , and , in terms of the , t/^ , and . = A; f / y / - ^ ' - z^z/ - ^  7 7 ^ ^ ^ ^ ^ // ^  /z ^ ? JZ/ ' f/ ^ ^ ^ y Z / ^ ^ Z / ^ Z / ^ ^ ^ HH', - -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 ^  ^6/77^^- K ^ j 7/77'/; / m " - 36 ^ ^ /77 - J J 6 6/77 V -^J/77^, ^ 6 777 ^  777777)-5 777777^ y-^ 777 / / J/77S - 777 * ^ 35 12. Bibliography. 1. L. E. Dickson - Algebras and Their Arithmetics. The University of Chicago Iress, Chicago, Illinois. 1923. 2. F. S. Nowlan - 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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