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The determination of sets of integral elements for certain rational division algebras Webber, G. Cuthbert 1932

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THE DETERMINATION OF SETS FOR CERTAIN RATIONAL b y G. C u t h b e r t OF INTEGRAL ELEMENTS DIVISION ALGEBRAS Webber SCBBHTESaMKS,. CAT. A<^j. NO. r^Jj-AZ.^. THE DETERMINATION OF SETS OF INTEGRAL ELEMENTS FOR CERTAIN RATIONAL DIVISION ALGEBRAS by G. Guthbert Webber A Thesis submitted for the Degree of MASTER OF ARTS i n the Department of MATHEMATICS The Un i v e r s i t y of B r i t i s h Columbia A p r i l - 193E TABLE OP C C K T M T S 1. Introduction. 2. Synopsis of Results and Formulae, Obtained by H u l l , whioh are Required i n t h i s Paper. 3. The Solution of Congruences (12)(mod 9) and (13)(mod 27). 4. Case I. 5. Case I I . 6. The Maximal!ty of Sets I(° and . 7. Case I I I . 8. Case IV. 9. Cases V, VI, and VII. 10. The Maximality of Sets if , I,0 , I? , , E®, rr-i-o and 11, , 11. General Case, S - ^ € 12. Conclusion. 13. Bibliography. 3 1 THE DETERMINATION OF SETS OF INTEGRAL ELEMENTS POR CERTAIN RATIONAL DIYISIQK ALGEBRAS 1* Introduction. The purpose of t h i s paper i s to determine i n t e g r a l elements of a ce r t a i n associative d i v i s i o n algebra 1, D, of order nine over the f i e l d of. r a t i o n a l numbers. The nine basal unit3 of D are y'' (^J - o, where with & a r a t i o n a l integer having a r a t i o n a l prime factor of the form <1 sA -2" 2. or £ I* that does not occur to a power whioh i s a multiple of 3. Also, x, s a t i s f i e s the cubic (1) Z* - 2>Z f-1 - O which i s c y c l i c , i . e . , i t i s i r r e d u c i b l e and has roots ~$. , f ' , ? ' , of the form 2 Further, where x ' - • This i s a special case of the algebra of order ^ over a general f i e l d F, discovered by Dickson, and ca l l e d by Wedderburn a Dickson Algebra. See Dickson's Algebras and Their Arithmetics, p. 66. F . S. Nowlan - B u l l e t i n of the American Mathematical Society, V o l . 3E, p. 375 (1926). 2 I t follows that X = and x = 6Cx-') = & Cx) s a t i s f y (l) , and thatz* e "Y-*) . . Integers of the cubic number f i e l d , /< O ) * defined by a root of (i) are of the form (xj z. + t with .1, , y., i and ^ r a t i o n a l integers. Professor F. S. Nowlan 1 has shown that the norm of a prime of U Cx) , not as-sociated with a r a t i o n a l prime i s either 3 , or a r a t i o n a l prime of the form <?^t/. In case the prime i s associated with a r a t i o n a l prime, the norm i s the cube of the r a t i o n a l prime and i s thus of the form £ /. Rational primes other than 3 » or those of the form ± I , are primes of K (*) . Further, every r a t i o n a l prime %Ji *• I i s factorable into three conjugate primes of , and so i s the norm of a prime of / < ( * ) . Thus the r e s t r i c t i o n on £ insures that D i s a d i v i -p s i on algebra . In future developments, we s h a l l need a c o r o l l a r y to the above, v i z . , the form 2*, given by F = 3 + £ A - 3 JL0 JL, % 3 JL„ ^  ^ + ?K -1 F. S. Eowlan - B u l l e t i n of the American Mathematical Society, V o l . 32, p. 379 (1926). 2 Algebras and Their Arithmetics, p. 68. 3 of the norm N(z-) of the general integer, (x) , of K-(x) repre-sents / , 3 , a l l primes of the form i / and a l l pro-ducts whose prime factors are 3 and primes ±- I . The form represents no prime other than these. Other primes are d i v i s o r s of the form only when they divide each of Ji0 , /, j and J-j_ , and then they appear i n powers which are multiples' of 3 . The general element z. of D s a t i s f i e s a rank equation 1 v i a . , a certain oubic equation having unity for the c o e f f i -cient of the term of t h i r d degree and having i t s other co-e f f i c i e n t s r a t i o n a l i n t e g r a l functions of the coordinates of 2 . Moreover, t h i s element does not s a t i s f y a l i k e equa-t i o n of lower degree. 2 The i n t e g r a l elements of an algebra are defined as those elements belonging to some one set possessing the four following properties: R frank): For every element of the set, the c o e f f i c i e n t s i n the rank equation are r a t i o n a l integers. G (closure): The set i s closed under addition, subtrac-t i o n , and m u l t i p l i c a t i o n . U (unity) : The set contains the modulus 1. M (maximal): The set i s maximal, i . e . , i t i s not con-1 Algebras and Their Arithmetics, p. 1 1 1 . 2 Algebras and Their Arithmetics, p. 141. 4 tained i n a larger set having properties R, C , U. We r e s t r i c t the inv e s t i g a t i o n to those sets which contain the nine basal units /xf" ^ » We write S- £ where ^ con-tains only prime factors 3 or those of the form <?^k £ t , while £ has only prime factors of the form — X. or q A — IS t at lea s t one of which oocurs to a power which i s not a multiple of 3 ; and further, £ i s of the form^ C ^ / . This paper considers the problem when S - £ , £ having prime factors only of the forms RA — 2_ or 9 A ~ IS , at leas t one of whioh occurs to a power which i s not a multiple of 3 , but £ i t s e l f i s of the forms 9 A . H u l l * consi-ders, and treats completely, the problem when <f - £ i s of the forms 9A • ± x and 9A -£• IS , and also when cP = ^  £ , with £ r e s t r i c t e d as above. Hul l gives i n h i s the s i s a f u l l outline of the problem as i t pertains to the general algebra G. Certain r e s u l t s and formulae which he obtained are needed for t h i s paper and w i l l be l i s t e d i n the next section. x H u l l - The Determination of Sets of Integral Elements for Certain Rational Division. Algebras. M. A. Thesis i n Library of the University of B r i t i s h Columbia. 5 E. Synopsis of Results ana Formulae Obtained by H u l l , wnioh are Required i n t h i s Paper. The rank equation of T. f o r the algebra D i s as f o l -lows (Hull, p. 7 ) : ^p.y. Y +t^fl.Y +cJ-,fl.Y-i-''L'(3*y> • ft*K " 3 ft. Y +3 0^ o„Y> + 3 KKY^ -- 0 A oongruenoe of the form y i e l d s , on manipulation (Hull, pp. IS - 13): I X- £y = £.J - f s <y- X £ 0 * ^ 3 ) , 1 x 2 . \3-The c o e f f i c i e n t s of a*1" i n the rank equations of y"z.K Cc,S-°,i,^ y i e l d (Hull, pp. 9 & 14) : On solving equations (7) for the coordinates of z , we obtain (Hull, p. 9): «) < - x v ; +. v X ; 7 r IcP ya r u W , - a- V i / ( - u ^ (10) I qcT ft = -xW, a- V , f W , , The following combinations simplify the reduction of the necessary conditions that the rank property holds fHull, p.15): f si s 7c, -f- tcx ^ - iri + v~z ^ jt - to-, •+ -Wi-j -U-.+ U , | si.' - -Voi. , yt'-- + ^ , Substitution of f l l ) i n the c o e f f i c i e n t of u) and in. UlA as obtained from the rank equation f o r z: , y i e l d s the following as necessary conditions that z has the rank pro-perty (Hull, pp. 15 - 16): -f> x f ^ - 7- -h ?> s^'/t - st' and ^3 -^/LA-'p -1A-A-' -j-cI^A.A,'p -f> 1 8 Congruences (12,) and (13) y i e l d the following r e l a t i o n s (Hull, pp. 16 - 17): A.' -h € yi- '-h X ' = O (^utrzL -b) t - 3 ^  - er } 3i The Solution of Oongruenoes  (ia)fmod 9) and (15)(mod £7), Using the r e s u l t s of the l a s t section, we determine the necessary and s u f f i c i e n t conditions that "2- may have the rank property. Substituting (17) i n (12), and grouping, we require (12) - -VCr^ (X ZyiL. - ->yJ) - £.V~T_ (sy*, - £) = O {^T^o-d- 7>) . We obtain the c o e f f i c i e n t of CO i n the rank equation for 2. X from the corresponding c o e f f i c i e n t i n the rank equation for z. 1 . The condition that i t be i n t e g r a l i s as follows: (2.0) Substituting (17) i n (19), and grouping, we require - £.A.jt + C/y KS-x + L yt irx = a (^->^neL 3) . On subtracting, (18) — ( 2 0 ) , the following i s required: Introducing the notation t h i s l a s t congruence becomes For the necessary substitutions, see H u l l , p. 8 10 (2/J /yy^ Y ~ X - Yx = o (s^^i 1). Congruence (El) has the following solutions: r /Vw, ~ 0} Y — O ^ X / M ^ t ^ w u ^ (s*-*-wL 3) t Y = 0 ^ X 5 O (yy^tr-zL^) ^ ^-/^ Y = I X = 0 (yy^^^Ji T>) , Y = , X 1 I (yyy^A -b) f Y ? 0 . X £ ^ L ' V = i X = -I £y**r^L 1): Substitution of (17) i n (13)(mod 27) y i e l d s a set of so-l u t i o n s , eaoh of which i s included i n ( 2 2 ) . The following d i v i s i o n s of the problem are suggested by (22) : Case I. /yy, - o} Y~ o ^ X a r b i t r a r y (mod 3 ) , Case I I . STK. - 1, Y 3- 0 , X ~ o (mod 3) , Case I I I . Y = I, Y ~ 0 (mod 3 ) , Case IV. /».-/, X = / (mod 3 ) , Case V. ^ - Y£. 0 ) X ~ o (mod 3 ) , Case VI. ^ - Y £ X ~ o (mod 3 ) , Case VII. - -i, Y =  1, X = -I (mod 3 ) . In subsequent work, sub-oases w i l l be denoted by sub-s c r i p t s , e.g., 1^ meaning case I, sub-case 2; and sets of elements w i l l be denoted by 1- meaning the set contained i n sub-case Jj f o r which £ = j (mod 9 ) . Since £ = - / (yyu^-ti 1) t 12 4. Case I The conditions characterizing t h i s case are as follows: (2»; p = Lo- s y = a (mod 3 ) , -/- £ -f being a r b i t r a r y , (mod 3 ) . We make the following transformations, s a t i s f y i n g (23) and ( 2 4 ) , on /u , A, , At- t A' T A-' , and .X' : fait) { AU = 3 A, -tZT~ , V ~ - £ *~ , X ^ IX, X ^ , A' - 1st,' . The Ti. used i n (25) corresponds to the 7*- used i n Hull's transformations (59) , p. 2,0. Substitute (25) i n (13)(mod 8 1 ) , obtaining 3 [ (&, V € '+ X/) ^ + X '+ £*.,'+ X,')X 3 X + + i J*-, + X,) +7" L^c + Z + AT ,) - s>+~ [A, A,' A, '+A>,X/) -7K(A, 't*.^ +AT X, '-urj) {Zb) +y^iA,^i- A, v-^ + X, i^J) -(A\,+A, ') (^-S- £ v i - £(>,+ A,% iTf.1- 1+> ^ - [X , £ 7^ l/v) £ A, yi., -f- £ A, A,' i- A, ' X, t-A, X/ + t A,'X + £ A X , ') In order that congruence (26) may hold, we require or 13 since = ^ t). From congruence (27), we obtain, using (5) and (6), We use (28), a f t e r factoring and regrouping, to give the following: Writing x , - x\ + x/ , and making use of (27) and (29), the congruence (26) reduces to Xi. (i + x L - y,L- Y) = <? (^^t -h). This y i e l d s solutions which suggest the following seven subdivisions of Case I : /•n- and Y, a r b i t r a r y , (mod 3) Y, = i , a r b i t r a r y , (mod 3) Y, 5 (mod 3 ) , s>u = o Yt B o} (mod 3) , ^ = -/ Y, = - / J ">t- a r b i t r a r y , (mod 3) Y, z l j (mod 3) , /yu ~ a Yt s o i (mod 3) , ^ = / We now express the coordinates of z. i n terms of the parameters SL , % /t , ^ ' , , , ^ , , , ILO , / Sub-case I, . Sub-case 1^, Sub-case I. , Sub-case I Sub-case I f Sub-case I t  w Sub-case I 7 Xz = o Xx = i X = W X , = -/ -/ 14 , and -urt , using (7), (8), (9), (10), and (11), and then grouping so that the substitution (3X) R - s*>, + A.,' , S * s*-, + A - / , T - X, + yt,' t i s advantageous, obtaining - , ^ - ^, + , , f« - 3 • =, yt' + T- ^ This substitution f o r the /-V, 3^ , and ^-V , y i e l d s f o r 2. : where has i n t e g r a l coordinates, and 15 Thus the elements of case I which have the rank property are obtained hy annexing to the set 3- of elements having in t e g r a l coordinates, the elements of the form z./ , given by (33), whose parameters s a t i s f y (31). Theorem I. For case I, the necessary and s u f f i c i e n t conditions that the general element 2. of the algebras D with cT = £ y and £ of the forms - I , s h a l l have the rank pro-perty, are that the congruences i n (31) s h a l l hold simulta-neously i n each sub-case. Sub-case I . Transformations (32) and r e l a t i o n s (31) require and (27) requires Choose which s a t i s f y (34), obtaining a p a r t i c u l a r value of , Multiply A on the right by the two conjugates of / + 7c , obtaining e f = 1 + A (-1 -x-hx1-). Consider £ = / (mod 9). Using the above value of £ , we have 16 where Multiplying f35) on the l e f t by x and on the ri g h t by 2.' , and then subtracting, we obtain (3 7) z A, ; I + A,*\ Multiply (37) on the l e f t by x. and substitute f o r xA, from (37), obtaining (3?J » ( - i f + A l x . ' z . The remainder of the m u l t i p l i c a t i o n table may be com-puted by sim i l a r methods, but the r e s u l t s are not needed here. Squaring (35), and using r e l a t i o n s (37) and (38), we ob-ta i n (3?) y\ i + A , ( x ' l x - x 1 ) + + 3%.*) , By means of (35) and (39), we now express z./ , given by (33), i n terms of A, and x. • We obtain 3 v f T A , l ( - 6 + ^ + With £ = / (mod 9), reference to (34) shows that z,' has been expressed, with i n t e g r a l coordinates, i n terms of A, and ^ . The rank equation f o r A , , obtained by replacing the d U., 3^ la, , and of (4) by t h e i r respective values i n 17 ( 3 5 ) , i s ~ ou^-ft/A. = o ^ ^A> being a r a t i o n a l integer. These r e s u l t s show that i n sub-case I, , f o r £ - IJL + I , we have a set of elements possessing the rank property and having the basis A, z , i = <? ,f) , containing the o r i g i -n al basal u n i t s . The existence of the basis shows that the set i s closed under addition, subtraction, and m u l t i p l i c a t i o n . The set i s maximal (to be proved l a t e r , see section 10). The set oontains the modulus 1. Hence, according to the d e f i n i -t i o n * we have a set of i n t e g r a l elements. Consider £ ~ -I (mod 9 ) . The conditions that the elements of set I s h a l l have the rank property are, for t h i s algebra, j ft - S T- T a « The general element of t h i s set I ^ , with the term con-t a i n i n g A removed to the set since i t has an i n t e g r a l c o e f f i c i e n t , i s J-I f , i n (40) and (41), we replace - S by S , - ^ by V i , and — y- by -y- , we obtain 18 where R + S + T = o {yyy^l 1) ( f n + ^ s " •*). We now have the general element and the required condi-tions that we had for E - vJi+t , and (42) may be expressed, with i n t e g r a l c o e f f i c i e n t s , i n terms of the basis elements obtained there. By making the inverse transformation to the one above, we have a basis for the set l f ' J . Thus, f o r £ = 1A -1 , we have if. = -i-h A^ [x-t x - x*) }if" - l-h At,(x-ix--x1') + A*(-ci + lx + 3x3-)> Ar- i (! + ? ) ( / + *) , A^(i-i *) + A^X-H +IX+ ->*")]. where and The rank equation f o r Ax i s uo^ - LO^ - Ji - o . By reasoning s i m i l a r to that used for £ - i- I , we have, f o r fc - -/ , a set of elements possessing the 19 properties of rank, closure, unity, and raaximality, and with the basis A*' x &,s -°, i, *•) • This set i s , thus, a set of i n t e g r a l elements according to the d e f i n i t i o n given i n the introduction. Sub-case I x ^ Transformations (32) and r e l a t i o n s (31) require, f o r t h i s sub-case, ( R -h £.S + T 3 / ?>) } with "K- a r b i t r a r y (mod 3). The set I i i s not maximal, being contained i n set 11 f (see section 6 f o r proof). From t h i s , the set 1^ i s contain-ed within the set II^~' J. Sub-case I , ^ The conditions required so that the elements of t h i s set may have the rank property are as follows: R T" 6 S + T = I {^rrtL Relation (33), under conditions (44), beoomes Choose s a t i s f y i n g (44), and substitute them i n (45), giving, as a p a r t i c u l a r value of 2., ' , Consider £ = / (mod 9 ) , Relations (44) beoome r R -h S -h T - j s o (<^<rzL 3) > Also, the above r e l a t i o n s beoome 3 = 1+ + y ( x +• X 1 ) from which (tfZ) y = - 3 + f C, ( 7 - *r -The rank equation f o r C, i s as follows: w^— co - AL = 0 Substituting these values of and i n (45), we ob-t a i n - [ W t C * r * v-„ £, - ur, C( f- / , * * * a * j Referring to (46), we see that the c o e f f i c i e n t s of 21 Ct ' x. (V, j = o , i1 %) are r a t i o n a l integers. Thus C, 'x'fcj*0* h form a hasis of the set of i n t e g r a l elements of the d i v i s i o n algebra D, with <f= £ r V JL + i Consider £ = -/ (mod 9 ) . Relations (44) become f li - S +T-l = 0 [yy^l 3) , On a change of notation i d e n t i c a l with that used i n sub-case I , , the following r e l a t i o n s are obtained from the cor-responding ones for £ = / (mod 9) of t h i s sub-case: where 3 ^ - l-h x -y(*+xl-)) and The rank equation f o r i s Referring to (49), we see that z,' i s expressed i n terms of w C^ji-0) h with r a t i o n a l i n t e g r a l coefficients. Thus Cj" C°\J 2 °> *"J form a basis of the set of integral elements of the division algebras D, with Sub-case I,,. Consider £ = / (mod 9) . The following are necessary and sufficient conditions that the set 1 ^ of elements of the algebra D, with S-L= °isA+l shall have the rank property: We shall show that this set, 1® , of elements i s con-tained within the set 1 ^ of integral elements. Form two distinct elements of by choosing two sets of values for R , S , and T , say R, , S, , Tt and Rxt 5j_ , 71 , and one set of values for ^  , v\ , and KTX t say t v-J % and %<rj% where -I which are obtained from (33) by putting =. o and values for parameters satisfying ( 4 9 ) . c R, -h 5, f- T, = / 3) and 23 D,* •lf(Rl + StT + r,y')(u.x*i-A') and Subtracting, and replacing by /3' , S, - Sx by S ' , T, - Tx by 7*' , we obtain an element £>; - 0^ which belongs i n 1 ^ by closure: Subtracting the r e l a t i o n s f51) we obtain R' T- S ' + T' (yyyuc-d -b) . Referring to r e l a t i o n s (46), we have or and therefore - -ur^ - / . Substituting these values of v-x and T/I i n (33), and t r a n s f e r r i n g the terms containing A-t and to the set $-we obtain Choosing o , we see that 24 i a an element of 1^ , and then by closure, subtracting t h i 3 from z,' , we see that J^'^r^(l-hy-+- Yjt^-f-**) i s an element of 1^ . From (50) we obtain the following: (R-i) + 5 +- T ~ o (yy^o-cL i)} ^ = + 3 A j Substituting these values i n (33-), and replacing/?-/ by /?3 , the general element f o r I % beoomes where ^ + S f ? = 0 (yyyua-J. -\) and i n which the terms containing ^A. , yt^ , and have been transferred to the set 3- . Since ~3 +• 5 y-* ^T*) (* + x * + , where R3 f S f T 5 <? ( ^ K ^ 3) ^I. i s a r b i t r a r y (yy>^- T>) , belong i n , the necessary and s u f f i c i e n t condition that the general element of 1% be contained within i f i s that -^(i-txx-hx1-) i--^C'^y-+y-j^x be contained within 1^ (from the property of closure), or 25 that t h i s element be expressible, with r a t i o n a l i n t e g r a l co-e f f i c i e n t s , i n terms of C and . Substituting (47) and (48) i n - k + **) iC<+ *r+ y \ ? * 4 , we obtain Thus the set 1% i s contained within the set 1^ and so i s not maximal. Consider £ = -/ (mod 9 ) . The necessary and s u f f i c i e n t conditions that the set 1^! of elements of the algebra D, with <f ~ C * <?J{ -/ , s h a l l have the rank property are the following: A - S f- T a / (^n^rzi i) j i c x = ~ Uj_ = wx i>) ^ yy^ ~ —I The general element of , obtained from ( 3 3 ) , i s of the form a f t e r the term containing has been transferred to the set S~ For the set 1^ , the general element and the necessary and s u f f i c i e n t conditions determining the rank property bear 26 The same r e l a t i o n to those of I3 as the general element and necessary and s u f f i c i e n t conditions of In- hear to T(') those of 1, • From t h i s we may show that where R^. - S^-f7~^ = O (yyy^rd 3) are elements of the set ^ • Thus the necessary and suf-f i c i e n t condition that the set ly be contained within the set I 3"" i s that he expressible, with 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 , i n terms of and X , the basis elements of I ^ • In terms of and x , t h i s l a t t e r element becomes X-HX^ + -x^x^CSCLt -x*)x\ Thus the set i s contained within the set I , ^ and so i s not maximal. Sub-cases I b- , I c . and I 7 ± The necessary and s u f f i c i e n t conditions that the sets of elements i n sub-oases I s- , I 6 , and 1 7 of the algebras D, with £ = - q Ji ± i t s h a l l have the rank property are as follows: R + £ S +- T 5 - / (p^^L -b) i 27 where a- and A- are equal to o , / f or z . The necessary and s u f f i c i e n t conditions that the sets of elements i n sub-cases I l t I,, and I ^ of the algebras D, with £ - £ r 9 ^ 4 / , s h a l l have the rank property are as follows: f R -t e S + T ^ i (^^u ->>) t where a and ^- have i d e n t i c a l l y the same values as i n (52). The general element z./ i s given by (33) for a l l sub-oases. I f we replace ( R - R , 1 ^ - ^ ^ i n (52), we obtain the conditions (53). Since t h i s substitution replaces any basis element, such as C, of sub-case 1^ , by i t s negative, we w i l l have - -f a*13- yl= C^*, for these sub-cases, where -y- = ^(x^,^ and cPC^j £,) are the sub-s t i t u t i o n s used i n 1 ^ . Any general element z,' which may be expressed i n terras of x and d, , with i n t e g r a l c o e f f i -cients, may be so expressed i n terms of x and - (3, . Thus the sets of elements obtained i n these sub-cases are c o i n c i -S8 dent with those obtained i n sub-oases I x , I } , and I ^ • Theorem I I . In case I there ex i s t two sets of i n t e g r a l elements for each of the algebras D, defined by S' - £ - % A -t-1 and S - £ = <? A -i . For £ - y^A -f- i , the elements of the two sets are formed by annexing elements of the form (33), and l i n e a r combinations of these elements, where the parameters s a t i s f y (34) and (46) respectively, to elements of the set $~ . For £ - -/ , the elements of the two sets are obtained by annexing elements of the form (33) and l i n e a r combinations of these elements, where the parameters s a t i s f y (40) and (49) respectively, to elements of the set 5. Case I I 29 The conditions characterizing t h i s case are the follow-ing: f U-^. + £ 1/y -f- ur^ 5 o (^rnsv-cL l)) (i>-«T) \ A = ZA = X ( W t / l) L p £ £ O- £ T" ^  / (<rK«rel l) • and as a consequence of the l a s t two yL ' = £s±;' = s&' (^rh«rJ- T>) Choose the following transformations on the parameters AL. t sd t A * A' t A' > and X' , so that conditions (55) are s a t i s f i e d : ( = 3^< + , A.'- 1> s^.'-t-0 , These transformations give for , <r , and 7- , f - 3 (si,* +• I - 3 R. -hi } (?D j V ^ 3(>,^;)f £ - 3 S + £ , L r = 3 (^t, + st;) + i ^ IT -HI T Substitute (56) and (57) i n (13)(mod 81), using (30) and (32), thus obtaining -i[(R + tS+T) ^ ( R \ S V r ^ c v V - V V + 0*. A-i ' + Mi' + X, X,') + % n<,(Z A , X, + i A, + A X, -h C A , At, ' +- C st, U, + £ 'jr, +- C- ' +Si/X, -hSL, s^,'-hCA-/Xl'-ht /L.UZ-h A/X/) - i ( LA, XT '+- C A, 'A, -I-EA/X, A, A, '+SL, 'X, + A, X, ') 30 - (t s4-f 'yt, ' + C y,, 'A., ' + yv, 'yt,') * / * . ( V f v ^ ^ V £ ^  st.,')i-rll-'h)(E'U.iA.l+€vi. yt, +wx A,,) + x(!l+-£$ -tT) (V.-^fJ From v t f ^ Tv-t = o(yy**>-<k 3) we obtain r ("U^  -h I - 3 C f v A ^  + ^ 4 ^ T - £ Vi.) = - 3 ( £ VY I^O. ^ ^ -f £ vj) (if <?) = 3 ( " (yisy^cL 9) , Using (29), (30), and (59), oongruenoe (58) becomes XL f - ^ Xi+X,''' + X.X,' ~^Y,XX (60) i. , i T- a^Xt Y, + x Y, ^ o (simJL 3) _ Substituting (56) and (57) i n (12), and using (30), we obtain as a necessary and s u f f i c i e n t condition that the co-e f f i c i e n t of i n the rank equation of z. be i n t e g r a l , Substituting (61) i n (60), congruence (60) becomes X"/ +• X, X, ' + 1 X/ 2 O (sy^nl \)% y The solutions of (62) are as follows: )Cj_ =_ 0 ^ X , yt^3~ ^CAy^X^A^, j (^yyi^-d. l)) X x = ±1 j X, 5 O i^rd 3) . 31 This suggests the following d i v i s i o n i n t o sub-oases: Sub-case II, . = o s V] (sy*^JL T) } Sub-oase II x . Xu =. I ~ Y t {yyn^-zi i) Sub-oase I I , . Xj_ =-/ = Y, (yy^A . Theorem I I I . Por oase I I , the necessary and s u f f i c i e n t conditions that elements of the algebras D, with <f - £ 9A. ± / , s h a l l have the rank property are given by the congruences characterizing the above sub-cases. Using (8), (9), and (10), with (11) and (32), we express the J-**. % f3'/±. , and -^U- i n terms of the parameters R , <5 , T , U-x t ir^ , j and , as follows: a - v- - i . S - A^-(4o ~ y * -b ° 1 > a - i S - n. , p, - 3 ^ <? , 32 The substitution of the above i n the expression for z. y i e l d s Z. = lc, - R +• R x + A., ' z.x + f{ " $ + 5 X -IS*-, 'z^} where -•3 ^ y ^ yJC** ^) Thus the sets of i n t e g r a l elements of oase II are formed by annexing to the set 3- of elements with i n t e g r a l coef-f i c i e n t s , the elements given by f63) t where the parameters obey the conditions required i n each sub-case, and l i n e a r combinations of such elements. Sub-ca3e II, ^ The necessary and s u f f i c i e n t conditions that a set of elements, given by (63), of the algebras D, with cT = € - <? A. £ / t s h a l l have the rank property are as follows: •U,^ = £ 2 IAT^ (yryiyrt^ 3 ) ? 33 being a r b i t r a r y Consider £ = / (mod 9 ) . With £ = qA+i , the conditions (64) become ' ft + S +• T B O (yyy^-el T>) v. yyx, being a r b i t r a r y (e^enL ^) • and the general element z./ t given by (63), a f t e r the terms containing A have been removed to 4- , becomes Prom (65) we obtain ( 0 7 ) i w ' A , T z R + A . + iA,, the expressions f o r 5 and 7 following from /? - S £ S -T ~ T- R = A, which, i n turn, follows from the f i r s t congruence of (65). Substitute (67) i n (66); we obtain tc?) ~i ^U+y+rj(x+ xj a f t e r the terms with i n t e g r a l c o e f f i c i e n t s have been removed to %~ . Substitute i n (66) the following values of the para-meters; 34 These values s a t i s f y f65). We then obtain as a spec i a l value of z,' , Multiplying H, on the ri g h t by the conjugates of + we obtain (jo) 1+ y + y - 3 #, ( -/ - x) . Squaring from (69), and using -y ? - £ we obtain We may take £ = lo , since IO i s of the form q^h+i and has prime factors of the form lA. +1 and <tsA + s~ , neither of which occurs to a power which i s a multiple of 3 . With £ - io t 1,7 H, x hecomes 3///= - x + x + YL 1 - *J, which, on right-handed m u l t i p l i c a t i o n by the conjugates of I - x*~ , y i e l d s {yi) Y ~ 2-' ^  + ^ ^Cu ~ *-^XJ. Substitute (71) i n (70), obtaining Substitution of (70), (71), ana (72) i n (68) y i e l d s , f o r the general element, z.,' v-3R.fl, x J ( j J - * t * j t ) / / ( * + M,x(x-xxl)\ -[^H,(i-*x-x^) + (st.Y+ ^y-yfr+xi •h U, (x - w- x. - x + /•*<- H, (a-3i)xrx. Thus z-,' i s expressible, with r a t i o n a l i n t e g r a l co-ordinates, i n terms of H, and x . (73) 35 The rank equation of //, i s We have, now, a set of elements, given by (66) where the parameters s a t i s f y ( 6 5 ) , of the algebra D with cf = £ = ? ^4. + i t which have the properties of rank, closure, unity, and maximality (to be proved l a t e r , see section 1 0 ) . By d e f i n i t i o n , t h i s set i s a set of i n t e g r a l elements with the basis H^ (V j ' ~ /, ^ ) . Consider £ = -/ (mod 9 ) . With £ = tJi - I , the conditions (64) become ( ft " 5 T- T £ O (yyy^rJL ^ - Vx + = O (sy^r-d. 3) _ c ^ being a r b i t r a r y -\). The general element z,' , given by ( 6 3 ) , a f t e r the terms containing have been removed to , becomes Replacing 5 Jry - S ) 1TX Jry " ^ , ^ ^ " 7> , (73) and (74) become (65) and (66) respectively. Making these replacements i n ( 6 9 ) , w i l l serve as a basis element f o r the set of elements i n t h i s sub-case, 36 We have, then, a aet of elements, given hy (74) where the parameters s a t i s f y (73) , of the algebra D with cT~ £ = fA -/ , which have the properties of rank, closure, unity, and maxi-mality (to be proved l a t e r , see section 1 0 ) . By d e f i n i t i o n t h i s i s a set of i n t e g r a l elements with the basis 7/^ ^ (sJ-",', x). Sub-oase I I , . The necessary and s u f f i c i e n t conditions that a aet of elements of the algebraa 3), with <T- z - <?sfi- ± I , s h a l l have the rank property are as follows: R + <£ 5 +• T ~ I (yy^trel -i)t + € + xu-^ = o U^<r-eL l) {75) < U^- tV-^= £V-X - Ur^_ = ic^ = / T>) t SYC being a r b i t r a r y Qvr^cXi) . Consider £ = i (mod 9 ) . With £ = 9 A •+ i , congruences (75) become C R + S + T = / (yy^rd 3) f(76) < V-^-f v«v s 0 (y^^c-cL -b) and the general element i s given by ( 6 6 ) . Erom ( 7 6 ) , with R-l - R' we obtain the following: R'~ 5 = 5 - T ~ T- R' = At ( W * i T>) T and therefore 5 . l\'-A, + iA^ T - R' 4-A, +*A, and also 37 ~ -ur^ -f- I + 3 J, } vc^ r -ur^ - I +• 3 Au . Substituting these i n the general element and tr a n s f e r -r i n g the terms containing ^ , ^ , ^  , and yix to the set 4- , r e l a t i o n (66) becomes (77) -\{y^{i+r + rx) + (-/ + y)](* + *y) In the same manner, the general element f o r the set 11^ may be written i n the form: the parameters s a t i s f y i n g (65). Taking x^ v = <? i n (78), and any set of values s a t i s -fying (65) for the remaining parameters, we obtain an element of II f . Taking IAT^ variable and the same set of values for the remaining parameters, we obtain another element of 11,^ , On subtracting these two elements, the element i s determined as being i n the set 11^ . With R - S - T « vw^ ^  >v = o i n (78), the element i s i n the set Il'f . Using an argument sim i l a r to that used for y^. ( / v_y. + -v^ )^ -tx3-) , we see that 38 i s also i n the set II, 0 J . With A = S ~ 7* - and then R - 5 - T - I , determine two elements of II f whioh, on subtraction, y i e l d i (R + SY-hTyJO-x+x*) where R 4- S 4-T so (y-n^-A'h) } as an element of II f . Since the term i(Ri-Sr+ TyJCi-x + x*) with R-hS -hT = I (WW. T>) t of (77), may be written as -3- [R. + $ r -^T yXi-^t^ +J^(i-v-hx") , where A+• S +-T = 0 {sv>^-cl i>) t ^ j _ a obvious, using the r e -sul t s of the l a s t paragraph, that the necessary and s u f f i c i e n t condition that the set 11^ be contained within the set I l f , i s that t h i s l a t t e r element be contained within IIf . Using (71), t h i s element becomes X^-hH^tl-x-h**) CO and so i s an element of I I , . Thus the set Ilf° i s contained within the set I I f and so i s not maximal. Consider £ 5 -/ (mod 9 ) . With £ = 1 -A - 1 r e l a t i o n s (75) become 1 R - 5 +• T = / ^erU 3) f yy^ being a r b i t r a r y [ A y ^ L -3) . The general element i s given by (74). 39 The replacement of 5 Jnf - S J ^ Ay - v-t_) <y Jy - y,, transforms f79) into ( 7 6 ) , (73) into ( 6 5 ) , and (74) into ( 6 6 ) . Thus the discussion of the sets and I I m a y he redu-ced to the discussion of sets I I 1 W and I X ^ r e s p e c t i v e l y . As a r e s u l t the set IIX° i s contained within the set II C7' J and so i s not maximal. Sub-case I I „ ^ The neoessary and s u f f i c i e n t conditions that a set of elements of the algebras D, with ef- £ - f^ft ± J , s h a l l have the rank property are that the following congruences s h a l l hold simultaneously: /? +- £ S T~ 7~ S -I (sywxl -b) , /n, being a r b i t r a r y (^y^ti %) . The general element i s given by ( 6 3 ) . Employing reasoning s i m i l a r to that used i n sub-case I I t , the necessary and s u f f i c i e n t condition that the set I I f be contained within the set I l f i s that be expressible, with r a t i o n a l i n t e g r a l coordinates, i n terms of U, and X . As seen i n sub-case I I t , t h i s i s possible. Therefore the set I I ( l J i s not maximal. S i m i l a r l y 1 1 ^ i s not maximal. 40 Theorem IV, For oaae I I , there e x i s t two sets, II and II , of in t e g r a l elements, one i n eaoh of the algebras D, defined by cT - i - tJi+t and o° - £ - f A -i r e s p e c t i v e l y . 41 6. The N on-Maximal i t y of Seta i f and I (I ' J ^ We deduced i n the l a a t aection that the following ele-ments were contained i n the set I I f : and so, adding a multiple of 3 , ^ ( " - y * (* + *-h where R ' +• S f - f = <3 3) ^ i ^ i and 'H- being a r b i t r a r y Q^^^l ->>). Since the general element for the set 1^ may be w r i t -ten i n the form and since the above elements are contained i n I I , W , i t i s necessary and s u f f i c i e n t to prove that i s an element of the set Il(' J i n order to prove that the set 11^ contains the set I f . Using (71), t h i s l a t t e r element becomes Thus the set I f i s contained within the set II f ; . 42 Since both IC~'J and Il|"° are obtained from i f and l i f respectively by replacing S by - S , v\_ by - t y , by - y t the set IC~'J i s contained within the set II^0 and so i s not maximal. 7. Case I I I . 43 This case i s characterized by r -f- £ ir^ + vwx = o ((vrx^rtL }) t ($}) I A.-t/>-$i.si--St = /t-/s.si ^yy^-JL'b)^ S a t i s f y i n g (81), choose the following transformations on the parameters A. , /X , st , A> ', A- and A' : f yi - 3/V,T-^H- , = 3 >*, ' - 6*- -') , ($2.) 1 yj, = 3AL,+L(^-I) , ^ IA.,'-e(<^+i), The substitution of (82) i n (13)(mod 81) y i e l d s , a f t e r using (29) and and the substitutions (30): /TV- Xj_ f T" /-x-X*. Y, -/- € v-j. X t Y, Substitution of (82) i n (12) y i e l d s as the only solution, and as a consequence R - £ S = £5-7- = 7" - £ 3). Combining (84) with (83), we obtain (g s) X^ [AT, -h ^  a l l of the solutions being included i n (84). 44 Theorem V. A comparison of (84) and (85) reveals that the necessary and s u f f i c i e n t conditions that the elements of the algebras 3), with cP = £ = £/ t whose parameters, given by (8£), s a t i s f y (81), have the rank property are as follows: ( R 4- L S i-T ^ O (^ruo-eL -7,) > and Tx- being a r b i t r a r y (^^L -*,) . Employing (7) - (11) and (8S), we obtain for the co-ordinates of the general element z i n t h i s case: / = "K- - 2=- Q _ 3s. o 3 * 1 ) I 3-ft - 'u">- + Jh~ A^. ^< i- 3 + 7 , n V - i - S f a. s _ # e_ 3" <T j The general element i s Z. = z, f z, , where z., ^  has r a t i o n a l i n t e g r a l coordinates, and ( ^ f * \ Y + TAT*, * 4 5 Thus the elements belonging i n t h i s ease and having the rank-property are obtained by annexing elements of the form (87) whose parameters s a t i s f y conditions (86) to the set of elements having i n t e g r a l coordinates. Consider £ s / (mod 9 ) . With £ = qJc-hi , the conditions (86) become R f S + r = o Q*y*^zi -i) t being a r b i t r a r y (yy^-d i) j where A i s o , / , or a. . The general element ' , aft e r t r a n s f e r r i n g the terms with i n t e g r a l c o e f f i c i e n t s to 0- , becomes It may e a s i l y be shown by the methods used i n former oases that the following elements, whose parameters s a t i s f y the conditions noted, are contained i n set I fJ : f--h yy^ a r b i t r a r y (/wW l) 46 From t h i s i t i s necessary and s u f f i c i e n t to show that i s an element of set I ^  i n order to show that the set I I I ^ i s contained i n i f . Substitute (35) i n t h i s element, ob-tai n i n g A, i-i + z) +- A, . Since the general element for the set of elements I I I ^ having the rank property i s expressible, with r a t i o n a l i n t e -gral coordinates, i n terms of the basis elements of set i f , the set I I I W i s contained within the set i f , and so i s not maximal. Consider £ = -/ (mod 9), With £ = <?A -i , the conditions (86) become R - 5 + T 3. a Uyy^L ) and the general element becomes ~3" ( ^  + V\. y- + i^vy-^ O +• x u) (l-y + y1-)C-3- + %x-hurx') --L Replacing the discussion of sets I(~'J and III ("^ may be reduced to the discussion of sets i f and III ( , J respectively. Thus the set I I I ^ i s contained within the set l\]) and so i s not maximal. 47 Theorem VI. There exi s t no sets of i n t e g r a l elements i n either alge-bra D i n oase I I I . The elements, belonging i n t h i s case, which have the rank property occur i n the i n t e g r a l sets i f and 1 ^ . 48 8. Case IV. The conditions characterizing t h i s case are as follows: ($0) < A. -£ ~ ZA.-yt E jt-y\ = -1 (^rvl i ) j L p = £ a- ~ T = / t^y^cl -3) _ Choose the following transformations on the va r i a b l e s A, , / i i A,'* A'* and -z^ such that conditions (90) are s a t i s f i e d : f si = 1SLt-f--h. t A-'- ?A,'-(^-I) J L At ~ ^7t, *•(-*>-!), = -bst,'- (-*-+') , Substituting IA.^ ~ T^ 3 -f / i n TA^ + e^f- ^ = /^ T <^.?J ) we obtain These transformations (91) necessitate the following: r p - + A.,') -t- 1 •= 3 R + l t L r ~ 3 + /*-/) 3 7 - 1. Substitute (91) and (92) i n (13)(mod 81), using (29), (30), and (32), and replacing we obtain 49 The substitution of (91) and (9S) i n (IS) y i e l d s as a necessary and s u f f i c i e n t condition that the c o e f f i c i e n t of cc i n the rank equation of "z. be i n t e g r a l : Combining t h i s expression with (93) , we obtain The set of solutions of (94), whioh contains a l l the solutions of (95), i s as follows: ( * a . « ~ ' j SK. + Yi_ =• o Q^^cL i) > X,. = 0 t + Yx. 2 - / ( ^ a J i -)) Theorem VII. For case IV, the necessary and s u f f i c i e n t conditions that elements of the algebras D, with <P - £ - R^A ± I , s h a l l have the rank property are given by (96). The coordinates of the general element z. f o r t h i s case are the following: Pi 3 ^ <? . a - A.' -H 5 - ^ - e , J _ y , = - - T T + ^r, «»- -^i + 3 • ? , 50 And i n terms of these, the paxt of the general element which has f r a c t i o n a l c o e f f i c i e n t s becomes ($7) - ^ ^ - j - f i ^ f ^ ' J -By substituting f o r y- and. y - i the r e l a t i o n s express-ing them i n terms of , the basis element of set i f , we may express z/ , given by (97) with £ - <?A. -hi , i n terms of and x with i n t e g r a l c o e f f i c i e n t s , f o r each of the sub-cases defined by (96). Theorem VIII. With t = 1 Ji + I t a l l the elements i n case IV having the rank property, given by (97) where the parameters s a t i s f y conditions (96), are to be found i n the set of i n t e -g r a l elements I 5 0 J . As a r e s u l t , case IV contains no sets of in t e g r a l elements for £. = 9 t . S i m i l a r l y i t may be shown that i t contains no such sets for £ - 9 - / 51 9. Oases V. VI, and VII. The oonditions characterizing cases V, VI, and VII, and the l o g i c a l substitutions to be used i n each case may be gen-er a l i z e d as follows: < yt - £yi~ = Z yt ~ A:-si. 5 -a, (yy^cL 7,) i L P E t ~ T ~ -I {yyt^r<l ?>) • |- / : 3 / t ( - X % yi, ' r 3S, V , 1 si, ~ 3AL, - £ ( ^ - ^ ) t ^' - 3 / 4 ( ' f £ ( / > u . / - ^ L st - 1st, - + °) > yt' - 3 ^ , ' + ~ I +<*>) t au , y^-, and being 0 , / , or x. • The oonditions characterizing cases II, III, and IV, and the substitutions used i n eaoh case may be generalized as follows: L ^ £ £ <T = T ~ I (yy^ci l) ; f 3/1 f - ^ j y t ' r 3 ^ , V ( / - V > , ^ yt - Isti-i-C^**) , yt' = 3ytt't (i-^-a.) t where , s5-t and "H- have the same values as above. The conditions (100) become conditions (98), and the substitutions (101) become (99) i f we make the following replacements i n (100) and (101) re s p e c t i v e l y : 5S ^ 3 - Ay - y 1V\ Jry - t<T^ } S*- , Ary - si,, ^ si, Ay- - , X ( " A, y A, whioh obviously necessitate that the following replacements be made: The same substitutions reduce the general element i n each of the oases V, VI, and VII to the general elements of I I , I I I , and IV respe c t i v e l y . Theorem IX. The sets of i n t e g r a l elements contained i n cases V, VI, and VII are i d e n t i c a l with the sets obtained i n cases I I , I I I , and IV. R Ay - R , T Arf -T . 53 10, The Maximality of Seta  i f , I<*>, 1% ,, i f , I I ? , and 1 1 ^ . The maximality of eaoh of the seta mentioned i n the heading i a not obvious. This i s due to the faot that the parameters, i n terms of which the necessary and s u f f i c i e n t conditions that the elements have the rank property are ex-pressed* are not independent, since si, , ALT A;,* AT ', A/> A/t are functions of the u U , v~A , and -urA • We determine the maximality of the above mentioned sets by expressing the basis element of each set i n terms of the basis elements of eaoh of the other sets. Substitute (47) in ( 3 6 ) , obtaining Thus A, i s not i n the set i f . S i m i l a r l y A, i s not i n the set II® , and so the set if i s maximal. In l i k e manner eaoh of the sets l(° and II/ J may be shown to be maximal, S i m i l a r l y , for £ = 9yk -I t we may prove that the sets 1 ^ , I ^  , and I I , are maximal. 54 1 1 . General Case, S - 7j £ « H u l l shows i n hia paper (Hull, pp. 28 - 31) that the necesaary and s u f f i c i e n t oonditiona that the elements of the algebras D, with cT = \ £ , where % i s the product of po s i t i v e i n t e g r a l powers of 3 and l i k e powers of r a t i o n a l primes of the form I i s h a l l have the rank property are the same as f o r the algebras D with cP = £ , but with respect to a new set of basal units given by f, x s - <v,^) where 7 - = y--£> being a number of KC*) » and ^f,3 - £• . In terms of these new basal u n i t s the (I^A and y^A. being r a t i o n a l i n t e g r a l functions of the o r i g i n a l coordinates of 2. , Since Hull does t h i s without taking into consideration the form of £ , his r e s u l t holds when £ i s of the forms q ± I , as i n t h i s paper. Theorem X. WOT the algebras B, with cT = \ £ , being the product of i n t e g r a l powers of 3 and l i k e powers of r a t i o n -a l primes of the forms 9A £ I , and £ i s the product of r a t i o n a l primes of the forms 9A ± 2- and 9 A - H- , 55 at l e a s t one of which occurs to a power not a multiple of 3 , hut £ i t s e l f i s of the forms - I t there exist sets of i n t e g r a l elements each of which corresponds to a set ob-tained for o° - £ , £- being r e s t r i c t e d as above, In eaoh case the set contains the o r i g i n a l basal units IS. Conclusion. 56 The following theorems sum up the r e s u l t s obtained i n t h i s paper: Theorem XI. Por the algebra D with cP = £- = <? A -h i , £ having r a t i o n a l prime factors of the forms lAi - 2_ , and 9 A ± Mr , at l e a s t one of whioh occurs to a power not a multiple of 3 , there exist three sets of i n t e g r a l elements, i f , 1^ , and I i f • The elements i n each of these sets are given by (32) f o r i f and I ? , and by (66) for I I f , the parameters s a t i s f y i n g the conditions (34) with £ » l(?>ucxl i)j (moa 9 ) , ( 4 6 ) , and ( 6 5 ) , respectively. Theorem XII. Por the algebra D with cf - £- = 9A -/ t £ being r e s t r i c t e d as i n Theorem XI, there e x i s t three sets of i n t e -g r a l elements, 1^"°, l f , J , and II<)~'J. The elements i n each of these sets are given by (41) f o r the f i r s t two sets, and by (74) for the t h i r d set, the parameters s a t i s f y i n g the condi-tions ( 4 0 ) , ( 4 9 ) , and (73) respectively. 13. Bibliography. I i L. E. Dickson 2. P. S. Ifowlan 3. P. S. Ifowlan 4. R. H u l l Algebras and Their Arithmetics. Un i v e r s i t y of Chicago Press, Chicago, I l l i n o i s . 1923. B u l l e t i n of the American Mathematical Society, V o l . XXXII, No. 4, July -August, 1926. Transactions of the Royal Society of Canada, Third Series, V o l . XXI, Section I I I , 1927. The Determination of Sets of Integral Elements for Certain Rational D i v i s i o n Algebras. M. A. Thesis, Un i v e r s i t y of B r i t i s h Columbia. 1930. 

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