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.
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- The determination of sets of integral elements for...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
The determination of sets of integral elements for certain rational division algebras Webber, G. Cuthbert 1932
pdf
Page Metadata
Item Metadata
Title | The determination of sets of integral elements for certain rational division algebras |
Creator |
Webber, G. Cuthbert |
Publisher | University of British Columbia |
Date Issued | 1932 |
Description | No abstract included. |
Subject |
Integrals |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-11-29 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0080409 |
URI | http://hdl.handle.net/2429/30181 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
Download
- Media
- 831-UBC_1932_A8 W3 D3.pdf [ 2.32MB ]
- Metadata
- JSON: 831-1.0080409.json
- JSON-LD: 831-1.0080409-ld.json
- RDF/XML (Pretty): 831-1.0080409-rdf.xml
- RDF/JSON: 831-1.0080409-rdf.json
- Turtle: 831-1.0080409-turtle.txt
- N-Triples: 831-1.0080409-rdf-ntriples.txt
- Original Record: 831-1.0080409-source.json
- Full Text
- 831-1.0080409-fulltext.txt
- Citation
- 831-1.0080409.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080409/manifest