2HK Wavmi&I'EC OF CHXCAGfO BOH-GYGLXC £X& M 3m& WLTB PUHE MAXIMAL SUBFIELB8 a m B s m r M n m . .s u b m itte d TBE FACULTY OF THE U1VXS10I? OF m tq FHYSXCAL SCI12KC I B CANDIDACY FOR TEE DEGREE OF MASTER OF SCIHHCK IXEPARTMEHT OF ISAaSIKSAti5!CS m urn umxL&cB. CHICAGO, XLLXnOXS MAH Gil, 1940 1 1 0 8 - - C Y C L I C ALG-EI3R tm.r, M A X I M A L STJBFIKLZJS 1 In & recent; paper' A- A. Albert proved the falsity of the converse of the well-known proposition that a cyclic normal division algebra contains a quantity J whose minimum equation is x a « J in the base field of the algebra* The proof consists of giving an ©xsaspl© of a non-eyelie normal division algebra con taining a quantity j as described above* The algebra described i n that example was of degree and exponent four* It is the pur pose of this paper to show that the exponent does not affect the property* and this we shall do b y cons trueting similarly am algebra of degree four whose exponent Is two. bo shall actually prove the following theorems • THEOREM. Let i and T\ be independen t ind e t e K i n a to a over the field R of real numbers > K ** E ( f # 7) ) • Then there exist non-eyelie norta&l division algebras, of degree four and exponent two over K* oaeh algebra having jaaubfieid K{t) of degree four over E such that t^ - V in K. To sake our proof wo use the known property stating: -that a normal division algebra of degree four has exponent two if and ■only if it is expressible as a direct product of two algebras of o degree two.Therefore we may take our desired algebra A to be A. Albert* Hon-oyellc algebras with pure maximal subfields* Bulletin of the American Mathematical Society* vol. 44 (l938)* pp. 67G-S79. ^A* A* Albert* IIoxvaal division algebras of degree four* Transactions of the American *Mbi3bMatical Society* vol. ‘ 8 4 '{1932}* p. 309* Theorem 6. -1- of the form (1 ) ft X A C « (1 , % t is ± v X Cl* -X* y, xy), whore the e*u 1tl plicafcl on tables of B and C are given by (2) 31 ss -ij* . Is *s u, 3® 88 «• (u 3^ 0, a ■/ 0 in KJ, yx « -xy* x® a v* yo » b (v / O, b / 0 in K). Wo aim'll make our proof by proper choice of the parameters u* vand -a* and tion » Y shallseek first In K* a quantity twith nijslsstus equa If we take t a a^i *• 8gJ agi} whore a^» Og, 8g aro In E(x) * then (3) t2 where e,‘ — a,2aa a t,•<*» -*• Iws* JL'u •¥• «o2a *•> *0 and bg aro in K* (4) (bx + bgx)2 « do desire tliac « Y t- gb-jbgX -i- b22v « V in 1C, that is* in K, while t/e v/lsla t2 not in K* that la, (5) bg ^ 0* Evidently (£} Implies that 2bjbg » 0 so that by (5) wo have b-^ * 0« We put ^1 ^ cl'j ■> b2x, ^'8 r” ^1 1 px * and have {8} l)^ ? bgx » (e^2 i- Se^CgX *- Cg2v)u * ( * 2d^dgX ^ as2v)a - {f]_2 f Sf-j^fgX + f22v}ua. Then b^ = 0 will be satisfied If and only If we choose u, v, and a so that (7) i cg2uv i d^2a + d2 Sva - f ^ u a - f22vua « O. To satisfy (V) we solve for a and obtain (8) a ~ - --d a,s c*”u * c^^uv , t*3®v - £ xsu • fo^uv ■ kikewis© (6) implies that wo satisfy {5} by choic© of our para mo tors so that C9 f b^ So^o^u. Bo^d^a — 2f.^f2us ^ 0 « We have now shown that for all values of u ■/ 0, v / G* cl* c2* ^l* dg, f^ and fg satisfying; (8) and (9), the algebra A defined by (X) and (2) contains a quantity t such that t^ « Y In K, t2 not in K. Xn proving A a non-eye11c division algebra wo uso tho O f} 3 2 following raet-ir.ocU Lot ~ r «# ft in K* S and 6 in E with L T {z} a quadratic field over K* that iko * A X. A Is a division algebra. be desire first to prove Tlxe algebras Bo « B X L and Co 33 C X 1* over L are genexmlised quaternion algebras over thotr reference field L. product & 0 = B 0 X € 0 * Furthermore A© ~ A X I* Is the direct Then it la Imowa^ that a necessary and ■ sufficient condition that A a over L shall b© a division algebra Is that the quadratic form {10} Q « f aAg2 * uuAg2 - { v A 4S -r bA52 - vbAp2 ) In til© variables A^, Ap, A^, * * ** A g In L shall not vanish for any A q not all zero in L . If 3 = Rft, 113 Is the integral domain of all polynomials in % and Y| with real coefficients, it Is obvious that, we may* °Tris Is the device used by A. A* Albert in bis psqjer, A o p a s time tion of non-eyelie normal division algebras * Bulletin, of the Ap©i*ioan Mathematical society, vol. 30 (1932), pp. 452453. 4 A. A. Albert, On the W e d d e r b u m norm condition for cyclic algebras, Ibid., vol. 37 (1931), p. 311, Theorem 3. without losa of generality, take the A^ In (10) to be in J £ kJ . Hence wo may write <11} A i => where tho 4 ^ and 80 ^ - are in J* ►iS (1 « 1, ***, 6), Than A^2 =* -»• * if (IS) P1 =rf12 + , 1S , Qt - 8 ^ ^ , the equation h -- 0 becomes (13) U?1 '■ ai'2 ** uap3 - VP4 * bp5 •t- (uQx But 1 and z are ‘ vbp6 ai---2 “ Uiv;y-; “ vQ4 - bQg j- vbhG js ~ O* linearly independent**with respect to il so that (13) Implies that (14) ) « uP-^ + al*g In (0) we a ~= let - e2 trnPg - vP4 - dj_ « 1, d2 « b P g -(- v b F g -- Q* = f2 « 0 and obtain ~(u -•uv)•Hot© that these choices also satisfy (1). Sub stituting this value for a In (14) we have f(*,)|) * uCP* ~ **) “ W P 2 4- u 2P5 (lb) _ -i- U^VPg - VP4 - bpg -** V b P g . Every quantity g ~ e(i» H > of J » h£I »)) 1 has a highest power £ n of ( with coefficient in ill I not identically zero. We shall call n the $ -degree of g and the coefficient of tho i “-leading coefficient of g. J[ n Similarly g has an 1f -degree arid an 7) -leading coefficient* p o 2 How we have assumed that 2 = 6 ■{-£**» A so that ®It Is obvious that we can have this condition hold by a suitable choice of 6 and 6 . -S- PI ; (1*1 *)^ ’ ^ i 6^ * t -degree? • (16) In fact* j;^ = Pjt2 ^1 + S1 (|*n ), where the degree of 2^ In (17) *I■ ’ ► ± in «T* must hi,vo even £%. = 34(1,11) in J, t is less than S ^ i p1 » T ±y\2Sf » c^Of), and q* * %.(>!) In Fpul, wiier© q^ lias degree less than 2#^ in 71 j and (18) T ^ i? 0* =* 0 if mid. only if P^ a A^ « 0* V;itli the above notations and conventions wo now arrange each of the seven tense of (15) according to descending powers of f , w-nose cooff Ici write are polynomials In to des cending powers of highest power of 7| * Mow the total coefficient of tho f appearing in the seven fcemas of (15) is the stas of seven or fewer polynomials in zero. H which Is not identically Furthermo,ro tills term is to appear explicitly .because the A^ and therefore the highest power of f (19) 7) arranged uccordiig are not all zero* suppose then thut this' in (15) were un odu. power, he lot u have odd f -degree and J -loading coefficient u-i v have even f -degree and t -loading coefficient v^* b liavo even f-degree and f -loading coefficient b^« Then the highest (odd) power of / Ira (15) can appear only in J (20) U ^P1 " F2' * uvP2 he not© first that t--ls highest power of only In tho torn —uvPg and therefor© m a t f cannot apisear appear ©lther in uCP^ - ?o) alone or Ira both u( t ^ - ?2 ) and -uvPg . In discus sing these possibilities we shall consider the following three eases* -6- (i) 2 px > SfQ, i n ) Ps denote tho 2 p1 < Bflg, ( H i ) 2 f>x « 2 p2 . t -degrees In case (I), the an<i therefore U 1^X of Here the p l and and Pg respectively as in (16 ). { -loading coefficient of uCP^ - Pg) Is S the highest power of must appear also in -uvP0 in order that the coefficient of the power be aero. Thus a u» the total coefficient is (21) u l9l ~ u lvlP2 :is 0 ln 7), 4 O. This liaplios that (82) , - v’ip2 - O in Vi'e choose v-^ to have odd Y|. Y|—degree, v-jf% and since and Pg have oven 7J -dogroe by (17), (22) is impossible and theroforo 2pg« =» Similarly (ii) Implies that (do) —u^Pg — 'd^v-^pg » O in 1}, 4 09 Pg .4 Of or (24) -1 - v 1 =* O in y\. Since (24) cannot be time by our choice of v^, Suppose, then, that ( H i ) holds and suppose that tho highest power of J 8Pp. » f)g. is not In-uvPg • First Then this condition Implies that (25) , u l^pl “ ~s 0 in 5 a n d .therefore (26) Thus the p1 ~ Pg. $ -degree of u(P-^ - Pg) is u “ -*• 2 ^ whoro u" denotes the $ -degree of u» J-f then we take tho 1 -degree of v as v ‘‘ ‘ > 0, we have that u*f i > u '' '*■ v" + 2p^ by (86), and the fact tilat we have supposed the ulghest power of / to be only in -u(P^ - Pp) * -V- Since this inequality is false* our hypothesis cannot 'hold. Hone© suppose that the highest power of -uvfg. 1 is also in Then the total coefficient of tho highest power of $ (2V) Is U 1^>31 “ i52^ ” u lv lp2 - 0 in ">). Vie consider the highest power of 7) and suppose t*.v t it la even. Let u-> have even 71-degree and 71-leuciing coefficient u<,» v^ have odd t) -degree and 7J-loading coefficient Vg. (20) Then the highest power of T| appears only in u^(p^ - Pgj. before* no have three cases according as ^1 ** ^2* Cf^ anti ffg denote the cr^ > ff’g, -degree of As cr^ < 9 g, or and pg respectively as In (1»). Hon the first two cases are clearly Impossible since in those ceres the leading coefficients would be UgT^ and -UgTg, respectively, and could not be aero without T ^ and thus having Pj, B ° or P jj " °* or T g being zero In. the third case we have the leading coefficient (29) U2(T 1 ~ ^ ® °* and therefor© (SO) Tx = T2. Hence the highest 'power of J|-d©sreo of u^. y| is u^v" j- 2 « j_ where Is the In the proof of the impossibility of (2:3) we have already taken v ^ e > O and tisu© we have, by (30) and the hypothesis that the highest power of 7) appears only in u^Cp^ - P2 )» that u^' i- 202. > U X* + *’ Stfi* dine© this is false v/e have now shown that this final case also Is not possible. Therefore the highest power of 7] in (27) must be odd and appears only In UjV^pg and cannot have total coefficient zero. B© can thus conclude that the high©at power of $ In (15) la not an odd power. Horace tho highest power of S in £15) must be an evon powt'i1 and appears only in (31) U 2P.. o. u'V?,o - v?.4 - biv & v b r6 v• Tli© loading coefficients of the terns of (81) are respectively (52) *viP4* -1)^5* V X1>1 % * Arrange these expreasions (32) according to descending pow rs of 7) « (33) Then the respective leading terms aro 8 —. ^yt3 ~ ° « «» 2 4^) " " 1 » „ **’ * 8 _ 2 tfd-.-2ui "-i-v-i ~u0 v0T../»1 o J. 1 . ~ ~ i -* -u 2 < k ^ i v . — -o2T57l v" 1 * v2h2T6^ w +b, * 1 * how v-^”' has a trendy boon chosen odd in s..owing that 2 a 2^g» and we now take ---= 7) -degroti of b-j_* as even. Then if the highest power of 71 appearing in the total coefficient (which is s osao awn of the expressions in (33)), wore an odd power, it would appear only in some or oil of tho terms of .£340 H oy/ "VSl'V "v 2T4» ~b2T5* > 0 by (18) and we choose Ug > 0 , ?g > 0, bg > G. Then no possible sua of the expressions In (34) eon bo zero. Thus the highest power of 7) In the coefficient rae are discussing cannot be odd and most bo even. It is easily seen that this implies that (SB) u 2ST3 <• vsbsT e » 0. Since this is impossible by (10) end our choice of Vg above, we have arrived at the desired contradiction of (15). -9- olrse© (15) does not hold, rm to<wr that A 0 over T. Is a division algebra. Since A 0 « A X n is 0 division algebra over L, so is algebra A a division algebra over K* By its form it Is a normal division algebra of degree four over K. The proof of the non-eyclie character of A is exactly the /» same as fchofc given by A , A. Albert in a previous paper. His proof consists of showing that A contains no quadrat1c subfiald IX(a)# O a* O 8^ r> , 8> field C containing E(z) * C in K.P and lienee no cyclic qu&rfcic Aith this, we hove the desired theorem. ®A construction of non-cyclic normal division algebras» ibid., vol. 3 0 1 1 9 3 2 ) , p. 454. VITA Hoy Lubiscii was born In Chicago* Illinois, on February 5, 191V. He received his elementary school education in the Chicago public schools and attended Tlldon Technical High 3clmo2. in Chicago for a year and a h a l f . His parents then moved to Port Huron, t-iieiiigon, and he completed his secondary education there, being graduated from Port Huron High school in June, 1034. Tho following February lie entered vyoodtrow hi Is on Junior College in Chicago tmcrc ho studied for a your and a half. Ho entered the Httivcrslty of Chicago in October, 1936, and received the Bachelor of hcieneo degree in Juno, 1S38. ho began his graduate work at the University of Cldcago in the summer of 1938 and in tho fall he acjcured a position as assistant in physics at Wilson Junior College• while working he continued to study at the University of Chicago and also attended the summer session of 1930. The instructors unuer whom ho has taken graduate courses are Professors Barnard, Hickson, jpribin, LVerett, Graves, L ane , Logsclon, Lttrm, Mac Lane and Reid. Although he has not boon able to take may graduate courses under Professor Albert as yet, he is especially grateful to him for suggesting the topic for this thesis and for friendly criticism during its preparation. •10-

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