On Division Algebras of Degree 3 with Involution

JOURNAL OF ALGEBRA ARTICLE NO.

184, 1073]1081 Ž1996.

0299

On Division Algebras of Degree 3 with Involution Darrell E. Haile* Department of Mathematics, Indiana Uni¨ ersity, Bloomington, Indiana 47405

and Max-Albert Knus† Departement Mathematik, ETH-Zentrum, CH-8092 Zurich, Switzerland ¨ Communicated by Richard G. Swan Received August 31, 1995

Let D be a division algebra of degree 3 over its center K and let J be an involution of the second kind on D. Let F be the subfield of K of elements invariant under J, char F / 3. We present a simple proof of a theorem of A. Albert on the existence of a maximal subfield of D which is Galois over F with group S 3 and prove an analog for symmetric elements of Wedderburn’s Theorem on the splitting of the minimal polynomial of any element of D. These results are then applied to the theory of the Clifford algebra of a binary cubic form. Q 1996 Academic Press, Inc.

INTRODUCTION Let D be a division algebra of degree 3 over its center K and let J be an involution of the second kind on D. Let F be the subfield of K of elements invariant under J. We assume that char F / 3. In the first part of this paper we present a simple proof of Albert’s Theorem wA 2 x on the existence of a maximal subfield of D which is Galois over F with group S 3 . The first step is a construction of a subspace of elements u such that u 3 g F, inspired by a similar construction in wH 2 x Žfor algebras without *This research was done while the first author was a visitor at the ETH, Zurich. he would ¨ like to thank the Forschungsinstitut fur ¨ Mathematik for its hospitality. E-mail: haile@ucs. indiana.edu. † E-mail: [email protected]. 1073 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

1074

HAILE AND KNUS

involution.. This construction was used there to give a short elementary proof of Wedderburn’s Theorem wWx that central division algebras of degree 3 are cyclic. In fact the argument given here yields another proof of Wedderburn’s Theorem. In wWx Wedderburn, in preparation for his result on the cyclicity of algebras of degree 3, proves that if u is a noncentral element of D with minimal polynomial f Ž X . Žsay., then there is an element j g D= such that j 3 g K and f Ž X . s Ž X y jy2uj 2 .Ž X y jy1uj 1 .Ž X y u .. In Part 2 we prove an analogous theorem for the elements of D symmetric under J. In Part 3 we apply these results to the theory of Clifford algebras. We prove that every central simple algebra of degree 3 with involution of the second kind is a homomorphic image Žas algebra with involution. of the Clifford algebra of some binary cubic form with its canonical involution and then show how to classify these images for a given form.

ALBERT’S THEOREM Let D, K, F, and J be as above and let S s Ž D, J .qs  a g D < J Ž a. s a4 be the F-subspace of symmetric elements. Let PaŽ X . be the reduced characteristic polynomial of a g D, let Tr be the reduced trace, and N the reduced norm on D. By passing to the algebraic closure of K one easily verifies the following formula for PaŽ X .: Pa Ž X . s X 3 y Tr Ž a . X 2 q N Ž a . Tr Ž ay1 . X y N Ž a . 1. If a f K then PaŽ X . is irreducible and the minimal polynomial of a over K. It follows that PaJ Ž X . s PJ Ž a.Ž X .. In particular J commutes with Tr and N and if a g S, then PaŽ X . g F w X x. We begin with an analog for algebras with involution of the proposition, p. 317, in wH 2 x. Recall that if d g D=, then IntŽ d . denotes the inner automorphism determined by d, that is, IntŽ d .Ž x . s dxdy1 for all x g D. PROPOSITION 1.

Let L be a separable cubic extension of F contained in S.

Ž1. There exists d g S l D= such that TrŽ Ld . s 0. Ž2. For d as in Ž1., the space U s dy1 L l Ker Tr s  dy1 l N l g L and y1 TrŽ d l . s 04 is at least 2-dimensional o¨ er F and TrŽ u. s TrŽ uy1 . s 0 for all u g U l D=. Ž3. The map J9 s IntŽ dy1 .( J is an in¨ olution of the second kind on D and U is contained in Ž D, IntŽ dy1 .( J .q. Ž4. We ha¨ e u 3 s NŽ u. g F for all u g U.

DIVISION ALGEBRAS OF DEGREE 3

1075

Proof. The proof is very similar to the proof of the result of wH 2 x mentioned above. For any x g S, the form f Ž x .Ž l . s TrŽ lx . has values in F since J ŽTrŽ lx .. s TrŽ JxJl . s TrŽ xl . s TrŽ lx .. Thus we get an F-linear map S ª L*, x ¬ f Ž x .. Since dim F S ) dim F L, there is an element d as wanted. For Ž2., since the form l ¬ TrŽ dy1 l . has values in F, we obviously have dim F U G 2 and, by the choice of d, Tr Ž dy1 l . s Tr Ž ly1 d . s 0

for all dy1 l g U l D=.

To prove Ž3. note that J Ž dy1 . s dy1 . It follows that IntŽ dy1 .( J is an involution of the second kind on D and that the space of fixed elements under IntŽ dy1 .( J is dy1 S > U. Finally we prove Ž4.: Because the element u is fixed by the involution J9, the argument presented immediately preceding the statement of the proposition shows that PuŽ X . g F w X x. Using the explicit form of PuŽ X . given there it follows from Ž2. that u 3 s NŽ u. g F, as desired. Let Dw X x s D mF F w X x. For any j g D=, u g D, we have X y jyiuj i s jy1 yi Ž j X y ju . j i . Thus

Ž X y jy2uj 2 .Ž X y jy1uj 1 . Ž X y u . s jy3 Ž j X y ju . 3 .

Ž ).

We apply this formula to the elements j s w 1 and u s wy1 1 w 2 , where w 1 , w 2 g U are linearly independent over F. We obtain: 3 2 y1 Ž w1 X y w 2 . s w13 Ž X y wy2 1 u w 1 .Ž X y w 1 u w 1 . Ž X y u . . Ž )) .

LEMMA 2. Then

y1 y1 Let u 1 s u s wy1 1 w 2 , u 2 s w 1 u 1 w 1 , and u 3 s w 1 u 2 w 1 .

Ž1. IntŽ wy1 .Ž u i . s u iq1 , i mod 3, and 1 wy3 1 Ž w1 X y w 2 . s Ž X y u 3 . Ž X y u 2 . Ž X y u 1 . 3

is the reduced characteristic polynomial of u i , i s 1, 2, 3. 3 Ž2. TrŽ u i . s u 1 q u 2 q u 3 and NŽ u i . s u iq2 u iq1 u i s wy3 1 w2 . Ž3. For the in¨ olution J9 s IntŽ dy1 .( J, where d is as in Proposition 1, we ha¨ e J9Ž u 2 . s u 2 and J9Ž u 1 . s u 3 . Ž4. There exist w 1 , w 2 g U linearly independent o¨ er F such that . TrŽ wy1 1 w 2 s 0. For such a choice we ha¨ e u 1 q u 2 q u 3 s 0.

1076

HAILE AND KNUS

Proof. The first part of Ž1. is clear. By Proposition 1 Ž4., Ž w 1 a y w 2 . 3 g F for all a g F. Since the field F is infinite it follows that Ž w 1 X y w 2 . 3 g F w X x. Because u 1 is a root of the right-hand side of Ž)). we get the desired formula for its reduced characteristic polynomial. Thus TrŽ u 1 . s 3 yi u 3 q u 2 q u 1 and NŽ u 1 . s u 3 u 2 u 1 s wy3 1 w 2 . Conjugating with w 1 , i s 1, 2, gives the other formulae of Ž2.. The claims in Ž3. follow from Ž w 1w 2 w 1 ., u 1 s wy3 Ž w 12 w 2 ., and u 3 s wy3 Ž w 2 w 12 ., because J9 u 2 s wy3 1 1 1 fixes U by Proposition 1 Ž3.. Finally we check Ž4.. Let w 1 be a nonzero . element of U. The form x ¬ TrŽ wy1 1 x on U has values in F. Since U is at . least 2-dimensional, there exists w 2 / 0 g U with TrŽ wy1 1 w 2 s 0. Since Ž . Tr l s 3 l / 0 for l / 0 g F, w 1 and w 2 are linearly independent over F. It then follows from Ž2. that u 1 q u 2 q u 3 s 0. To prove Albert’s Theorem we begin with a separable cubic extension of F contained in S Žfor example the F-subalgebra generated by any noncentral element of S .. We then obtain a space U as in Proposition 1 and . choose linearly independent elements w 1 , w 2 g U with TrŽ wy1 1 w 2 s 0, as y1 in Lemma 2 Ž4.. We then let u 1 s u s wy1 w , u s w u w , and u3 s 1 2 2 1 1 1 wy1 u w as in Lemma 2. 1 2 1 y1 y1 . Ž . THEOREM 3. Let E s K Ž uy1 2 u 3 if u 2 u 3 f K or E s K u 2 if u 2 u 3 g K. Then E ; D is cyclic o¨ er K and is a Galois extension o¨ er F with group S 3 .

Ž y1 . Proof. Assume first that uy1 2 u 3 f K, so that dim K K u 2 u 3 s 3. Since y1 y1 y1 Int Ž wy1 1 .Ž u 2 u 3 . s u 3 u 1 s yu 3 Ž u 3 q u 2 .

s y1 y Ž uy1 2 u3.

y1

g K Ž uy1 2 u3. ,

. restricts to a K-automorphism r of K Ž uy1 . IntŽ wy1 1 2 u 3 . If r is the identity, 2 the element uy1 u satisfies the equation y q y q 1 s 0. The algebra D 2 3 being of degree 3, this implies that uy1 u g K, in contradiction to the 2 3 assumption. Thus r is nontrivial of order 3. We further have for the involution J9 given in Lemma 2 y1 y1 y1 y1 y1 J9 Ž uy1 2 u 3 . s u 1 u 2 s u 2 Ž u 2 u 1 . u 2 s yu 2 Ž 1 q u 2 u 3 . u 2 y1 . .( J9 satisfies J0 Ž uy1 . Ž so the involution J0 s IntŽ uy1 2 1 u3 s y 1 q u2 u3 y1 and so defines an automorphism of order 2 of K Ž u 2 u 3 .. To show that . and J0 generate a group isomorphic to S 3 , it suffices to verify IntŽ wy1 1 . s IntŽ wy2 .( J0. We check it on the generator uy1 that J0(Int Ž wy1 1 1 2 u3 y1 y1 y1 y1 y1 y1 J0(Int Ž wy1 1 .Ž u 2 u 3 . s Ž u 2 u 3 .Ž u 1 u 2 . s Ž u 1 u 2 .Ž u 2 u 3 . s u 1 u 3 ,

DIVISION ALGEBRAS OF DEGREE 3

1077

where the next to last equality follows from the fact that uy1 1 u2 s y1 .Ž uy1 . Ž y1 . IntŽ wy2 1 2 u 3 g K u 2 u 3 and so commutes with u 2 u 3 . On the other hand we have y1 y2 y1 y1 y1 Int Ž wy2 1 . ( J0 Ž u 2 u 3 . s Int Ž w 1 u 2 . ( J9 Ž u 2 u 3 . s u 1 u 3

as claimed. 3 Ž . Assume now that uy1 2 u 3 s y g K. By Lemma 2 2 , we must have y s 1. If y s 1, we get u 2 s u 3 s u 1 , a contradiction to u 2 q u 3 q u 1 s 0. Thus y g K is a primitive cubic root of 1. It follows from u 3 s yu 2 that u 1 s yu 3 and u 2 s yu 1. Thus NŽ u 2 . s u 1 u 3 u 2 s u 23 and u 23 g K. Since J9Ž u 2 . s u 2 , we even have u 23 g F. Further we deduce from J9Ž u 1 . s u 3 and u 1 s yu 3 that u 3 s J9Ž y .u 1 , so that J9Ž y . s y 2 and K s F Ž y .. Thus we have K Ž u 2 . s . to K Ž u 2 . is given by K Ž u 3 . s K Ž u 1 . and the restriction r of IntŽ wy1 1 u 2 ¬ u 3 s yu 2 . It is then easy to check that  J9, r 4 generates a group of automorphisms of K Ž u 2 . isomorphic to S 3 . Remarks. Ž1. The argument used in the proof of Theorem 3 is largely inspired by the very last part of Albert’s proof in wA 2 x. The use of Lemma 2 allows us to avoid most of the computations in the first part of his proof. Ž2. As we remarked in the introduction the analog of Proposition 1 given in wH 2 x is used to give a proof of Wedderburn’s Theorem on the cyclicity of a central division algebra of degree 3 Žwithout involution. over K. In fact, Albert’s Theorem also holds Žwith the same proof. for D of the form A = Aop , A a central division algebra over F with the twist involution, so that Wedderburn’s Theorem can be viewed a special case of Albert’s Theorem. Thus we get another elementary proof of Wedderburn’s Theorem. A similar remark applies to the next proposition.

WEDDERBURN FACTORIZATION OF SYMMETRIC ELEMENTS Let D, K, J, S, and F be as above. We want to show that one can obtain the full ‘‘symmetric’’ version of Wedderburn’s Factorization Theorem Ždescribed in the Introduction.. PROPOSITION 4. Let u g D y F with minimal polynomial f g F w X x. There is an in¨ olution J9 on D such that u g S9 s Ž D, J9.q and there is an element j g S9 such that j 3 g F= and f Ž X . s Ž X y jy2uj 2 .Ž X y jy1uj 1 . Ž X y u . .

1078

HAILE AND KNUS

Proof. We first show there is an involution fixing u . This is a special case of a result of Albert wA 1 , p. 157x. For completeness we provide a proof. The elements u and J Ž u . have the same minimal polynomials, and hence are conjugates in D. Let J Ž u . s IntŽ g .Ž u .. If g q J Ž g . / 0, then IntŽ g q J Ž g ..Ž u . s J Ž u . and so the involution J0 s IntŽŽ g q J Ž g ..y1 .( J fixes u . If g s yJ Ž g ., then J0 s IntŽ gy1 .( J fixes u . We proceed as in the proof of Proposition 1 to find a 2-dimensional subspace W of L s F Ž u . ; S0 s Ž D, J0 .q and d g S0 l D= such that y 3 g F for all y g dy1 W. Let Y s F q Fu ; L. We claim that there exists l g L such that lW s Y. In fact it was shown in wH 2 x that for any 2-dimensional subspaces U1 and U2 of L, there is l g L such that lU1 s U2 . Again for completeness we recall the argument: let f, g g L* be such that U1 s Ker f and U2 s Ker g. The claim then follows from the fact that L* is a 1-dimensional L-space Žthrough the operation Ž fl .Ž x . s f Ž lx ... So let l g L be such that lW s Y and let j s Ž ld .y1 ; then j Y s dy1 ly1 Y s dy1 W, so that x 3 g F for all x g j Y. As in the proof of Lemma 2 Ž1., it follows that Ž j X y ju . 3 g F w X x and hence, applying Ž)., that Ž X y jy2uj 2 .Ž X y jy1uj 1 .Ž X y u . is the reduced characteristic polynomial of u . It follows that f Ž X . s Ž X y jy2uj 2 .Ž X y jy1uj 1 .Ž X y u .. Moreover J9 s IntŽ l .) J0 fixes u and j .

CLIFFORD ALGEBRAS The foregoing results may be applied to the theory of Clifford algebras of binary cubic forms. Recall that if g Ž u1 , . . . , u m . is a form of degree d in m variables over a field F, then the Clifford algebra C g of g is the algebra F X 1 , . . . , X m 4 rI where F X 1 , . . . , X m 4 is the free algebra on m variables and I is the ideal generated by the set

 Ž a 1 X1 q ??? a m Xm . d y g Ž a 1 , . . . , a m . , a 1 , . . . , a m g F 4 Žsee Roby wRox, Revoy wRe1 x, Childs wCx.. If g is a binary cubic form it has been shown ŽHeerema wHex, Revoy wRe 2 x, Haile wH 1 x. that C g is an Azumaya algebra over its center Z and that Z is isomorphic to the affine coordinate ring F w E x of the elliptic curve E given by the equation S 2 s R 3 y 27d where d g F is the discriminant of g. In particular each simple image A of C g is of degree 3 over its center and the simple images with center F are in one-to-one correspondence with the F-rational points on E. This correspondence gives rise to a function from EŽ F ., the group of F-rational points on E, to B Ž F ., the Brauer group of F, and it is shown in Haile wH 1 x, that this map is a group homomorphism. Now let g Ž u, ¨ . be a binary cubic form over F. The free algebra F X, Y 4

1079

DIVISION ALGEBRAS OF DEGREE 3

admits a unique involution fixing X and Y and this involution preserves I. We let ) denote the induced involution on C g and call it the canonical involution on C g . PROPOSITION 5. Let A be a simple algebra of degree 3 with in¨ olution of the second kind ha¨ ing fixed field F. There is an in¨ olution J on A and a binary form g Ž u, ¨ . o¨ er F such that Ž A, J . is a homomorphic image of Ž C g , ).. Moreo¨ er, if A is a di¨ ision algebra and f Ž x . g F w x x is irreducible of degree 3 with a root u g A, then there is an element a g F= and an in¨ olution J on A such that Ž C g , ). maps onto Ž A, J ., where g Ž u, ¨ . s a¨ 3 f Ž ur¨ .. Proof. Let K denote the center of A. First assume A is a division algebra. By Proposition 4 there is an involution J9 such that J9Ž u . s u and an element j g A fixed by J9 such that j 3 s a g F= and f Ž X . s Ž X y jy2uj 2 .Ž X y jy1uj 1 . Ž X y u . s ay1 Ž j X y ju . . 3

Hence the binary cubic form g Ž u, ¨ . s a¨ 3 f Ž ur¨ . satisfies g Ž u, ¨ . s Ž u j q ¨ Žyju .. 3 g Aw u, ¨ x. If we let J s IntŽ j .( J9, then J Ž j . s j , J Ž ju . s ju . Hence the map X ¬ j , Y ¬ ju induces a homomorphism from Ž C g , ). onto Ž A, J .. If A s M3 Ž K ., let J9 s IntŽ u.(t with u s diagŽ1,Ž 01 10 .., t Ž x i j . s Ž x i j . t , where x ¬ x is conjugation in K and t is transpose. If K s F Ž b . with b 2 s a, let g Ž u, ¨ . s Ž u y ¨ .Ž u 2 y a¨ 2 .. There is a homomorphism f of Ž C g , *. onto Ž M3 Ž K ., J . sending

X to

ž

0 1 0

0 0 1

1 0 0

/

and Y to



0 1 1

0 0 b

yb 0 , 0

0

where J s IntŽ f Ž X ..) J9. If A is a simple image of C g arising from the maximum ideal m of Z then ) will induce an involution on A if and only if m* s m. Moreover, because the center of A is Zrm we see that the fixed field of ) on A is F if and only if m has residue degree 2. In fact we can describe such maximal ideals quite precisely: THEOREM 6. The simple images of C g on which ) induces an in¨ olution of the second kind with fixed field F are in one-to-one correspondence with the pairs of points Ž r, "s . on the cur¨ e S 2 s R 3 y 27d such that r g F, s f F. The center of the resulting algebra is F Ž s ., a quadratic extension of F. Proof. Let C g s F X, Y 4 rI. We use the results of the discussion begin-

1080

HAILE AND KNUS

ning Section 2 of wH 3 x. By making a linear change of variables we may assume that g Ž u, ¨ . s au 3 q 3cu¨ 2 q d¨ 3 in F w u, ¨ x. Let r s ac, 2 s s ad, t s yc 2 , and d s s 2 y rt s Ž a2 d 2r4. q ac 3 , the discriminant of g. Let L s F w'd , v x where v is a primitive third root of one. Let X s Ž'd q s . X y rY and Y s Ž'd y s . X q rY in C g mF L, the Clifford algebra of g over L. If we let m s YX y v XY and n s YX y v 2 XY, then the elements mn , 'd Ž m3 q n 3 . are in C g and Z s F w mn , 'd Ž m3 q n 3 .x. Moreover Z is isomorphic to the coordinate ring of E : S 2 s R 3 y 27d via the map

R¬

mn 4 r 2d

,

S¬

'd Ž m3 q n 3 . 16 r 3d 2

.

Now the map ) m 1 is the canonical involution on C g m L and an easy computation shows that m*m1 s yvn . It follows that mn in C g is fixed by ). Hence the action of ) on Z s F w E x is given by R* s R and S* s yS. Now let A be a simple image of C g on which ) induces an involution of the second kind with fixed field F. As we have seen A s C grmC g where m is a maximal ideal of Z of residue degree 2 such that m* s m. Let K s Zrm. The involution ) induces the nontrivial automorphism of K over F. Hence in the coordinate ring K w E x there are two maximal ideals lying over m in F w E x. These maximal ideals are given by K-rational points and are conjugate under ). If Ž r, s . is one such point we have seen that r* s r and s* s ys. Hence r g F, s f F, and K s F Ž s .. Conversely, if Ž r, s . is a point on E such that r g F, s f F then K s F Ž s . is a quadratic extension of F. Moreover Ž r, ys . is another point on E and these two points lie over the same maximal ideal m of F w E x. Clearly m* s m and m has residue degree 2, so, as we have seen, ) induces an involution of the second kind on C grmC g with fixed field F. COROLLARY 7. Let K s F Žg ., g 2 g F be a quadratic extension of F. The simple images of C g with center K on which ) induces an in¨ olution of the second kind with fixed field F are in one-to-one correspndence with the pairs Ž a , " b . of F-rational points on the elliptic cur¨ e S 2 s R 3 y 27Ž drg 3 .. Proof. By the theorem the simple images of C g with center K on which ) induces an involution of the second kind with fixed field F are in one-to-one correspondence with the pairs of points Ž r, "s . on the curve S 2 s R 3 y 27d such that r g F and F Ž s . s K. Since s 2 s r 3 y 27d g F, it follows that srg g F. But then the points Ž rrg 2 , "srg 3 . are F-rational and lie on S 2 s R 3 y 27Ž drg 3 .. Conversely if Ž a , " b . are F-rational points on S 2 s R 3 y 27Ž drg 3 ., then the points Ž ag 2 , " bg 3 . lie on S 2 s R 3 y 27d and satisfy ag 2 g F, F Ž bg 3 . s K.

DIVISION ALGEBRAS OF DEGREE 3

1081

REFERENCES wA 1 x wA 2 x wCx wH 1 x wH 2 x wH 3 x wHex wRe1 x wRe 2 x wRox wWx

A. A. Albert, ‘‘Structure of Algebras,’’ Colloquium Publ. No. 24, Amer. Math. Soc., New York, 1939. A. A. Albert, On involutorial associative division algebras, Scripta Math. 26 Ž1963., 309]316. L. Childs, Linearizing of n-ic forms and generalized Clifford algebras, Linear and Multilinear Algebra 5 Ž1978., 267]278. D. E. Haile, On the Clifford algebra of a binary cubic form, Amer. J. Math. 106 Ž1984., 1269]1280. D. E. Haile, A useful proposition for division algebras of small degree, Proc. Amer. Math. Soc. 106 Ž1989., 317]319. D. E. Haile, On Clifford algebras and conjugate splittings, J. Algebra 139 Ž1991., 322]335. N. Heerema, An algebra determined by a binary cubic form, Duke Math. J. 21 Ž1954., 423]444. Ph. Revoy, Algebres de Clifford et algebres exterieures, C. R. Acad. Sci. Paris Ser. ` ` ´ ´ I. Math. 277 Ž1973., 655]657. Ph. Revoy, Sur certaines algebres de Clifford, Comm. Algebra 11 Ž1983., 1877]1891. ` N. Roby, Algebres de Clifford des formes polynomes, C. R. Acad. Sci. Paris Ser. ` ˆ ´ I. Math. 268 Ž1969., 484]486. J. H. M. Wedderburn, On division algebras, Trans. Amer. Math. Soc. Ž1921., 129]135.