ALGEBRAS IN A REAL CLOSED

TOMOHIRO KAWAKAMI AND IKUMITSU NAGASAKI,

Abstract. We prove that a finite dimensional division algebra over a real closed field R is isomorphic to R, C = R(i) or H. It is a generalization of Frobenius theorem.

1. Introduction Let R be the field of real . A D over R is called a division algebra over R if D satisfies the following two conditions. (1) D is a . (2) For any a, b ∈ D, r ∈ R, r(ab) = (ra)b = a(rb). We say that a division algebra is finite dimensional if D is a finite dimensional vector space over R. Theorem 1.1 (Frobenius theorem [3]). A finite dimensional division algebra over R is isomorphic to R, the field C of complex numbers or the quoternions H. There is a reference of Frobenius theorem in [4]. We can consider real closed fields as a reasonable generalization of R. Some concrete examples of real closed fields except R are Ralg = {x ∈ R|x is algebraic over Q}, the real closure of R(x) (e.g. [1], [2]). We can define division algebras over a real closed field R as follows. Definition 1.2. A vector space D over a real closed field R is called a division algebra over R if D satisfies the following two conditions. (1) D is a division ring. (2) For any a, b ∈ D, r ∈ R, r(ab) = (ra)b = a(rb). Fact 1.3. (1) The characteristic of a real closed field is 0. (2) For any infinite cardinality κ, there exist 2κ many nonisomorphic real closed fields. We have the following theorem as a real closed field version of Theorem 1.1. Theorem 1.4. A finite dimensional division algebra D over a real closed field R is iso- morphic to R, C = R(i) or the quoternions H for R.

Acknowledgment. The first author would like to express his thanks to Kota Takeuchi.

2010 Mathematics Subject Classification. 03C64, 16K20. Keywords and P hrases. Division algebras, real closed fields, Frobenius theorem.

1 2 TOMOHIRO KAWAKAMI AND IKUMITSU NAGASAKI,

2. Proof of Theorem 1.4 P roof of T heorem 1.4. Since D is a division ring, 1 ∈ D. Let < 1 > denote the vector subspace spanned by 1. Then < 1 >= R · 1 is isomorphic to R. If D =< 1 >, then D is isomorphic to R. From now on we write < 1 > by R. If D 6= R, then there exists an element a ∈ D − R. For any x, y ∈ R + Ra, xy = yx by Condition 2 of Definition 1.2 of division algebras over R. Let F be a maximal vector subspace containing R + Ra such that for any x, y ∈ F , xy = yx. Let d ∈ D such that dx = xd for any x ∈ F . Then F 0 := F + Rd satisfies the condition for any x, y ∈ F 0, xy = yx. By maximality of F , d ∈ F . Claim 1. F is a field. P roof of Claim 1. Let x ∈ F such that x 6= 0. Since x ∈ D and D is a division ring, there exists x−1 ∈ D. For any y ∈ F , x−1y = x−1yxx−1 = x−1xyx−1 = yx−1. Since x−1 is commutative with every element y ∈ F , x−1 ∈ F . By Claim 1, F is a finite dimensional vector space over R and it is a field. Since R is a real closed field, C = R(i) = R[x]/(x2 + 1) is an algebraically closed field with dimR C = 2. Claim 2. F ∼= C. P roof of Claim 2. For any n ∈ N, an ∈ F . Since F is finite dimensional, there exists the smallest n such that 1, a, a2, . . . , an are linearly dependent. Thus there exist n r0, . . . , rn ∈ R such that r0 + r1a + ··· + rna = 0 and rn 6= 0. Since C is algebraically 0 n closed, there exists a solution α = r + r i ∈ C of r0 + r1z + ··· + rnz = 0. Thus R(a) is 0 α−r ∼ isomorphic to R(α). Then r 6= 0 because a ∈ F − R. Since i = r0 , R(α) = R(i) = C. Hence R(a) ∼= C. Therefore F ∼= C. If D = F , then D ∼= C. From now on we write F by C. Assume that D 6= C. Then D is a vector space over C whose scaler multiplication C × D → D, (c, d) 7→ cd. For any non-zero c in C, the map fc : D → D defined by fc(x) = xc is a C-linear map. The inverse of fc is −1 fc−1 : D → D, fc−1 (x) = xc . Thus fc is invertible. Take c = i. Then fi ◦ fi(x) = −x 2 2 for any x ∈ D. Let λ1, . . . , λk be eigenvalues of fi. Since fi = −idD, λi = −1, where idD denotes the identity map of D. Since C is algebraically closed and commutative, the + − eigenvalues of fi are i and −i. Let D (resp. D ) be the eigenspace of fi with respect to i (resp. −i). Claim 3. D = D+ ⊕ D−.

+ − 1 P roof of Claim 3. Since i 6= −i, D ∩ D = {0}. For any x ∈ D, x = 2 (x + ixi) + 1 1 1 1 1 1 1 2 (x − ixi). Since 2 (x + ixi)i = 2 (xi − ix) = 2 i( i xi − x) = 2 i(−ixi − x) = − 2 i(x + ixi), 1 − 1 + 2 (x + ixi) ∈ D . Similarly, 2 (x − ixi) ∈ D . Since D+ = {x ∈ D|xi = ix} = C, D = C ⊕ D−. If x, y ∈ D−, then xy ∈ D+ because xyi = x(−i)y = ixy. If x ∈ D−, y ∈ D+, then xy ∈ D−. Claim 4. If j ∈ D−, then j2 ∈ R and j2 < 0. DIVISION ALGEBRAS IN A REAL CLOSED FIELD 3

P roof of Claim 4. Since j ∈ D−, j2 ∈ D+ = C. Clearly j2 ∈ R(j). Assume that 0 0 x−s − x ∈ C∩R(j)−R. Then x = s+s i and s 6= 0. Since i = s0 and x ∈ R(j), i ∈ R(j) ⊂ D . It contradicts the fact that D+ ∩ D− = {0} because i ∈ D+. Thus C ∩ R(j) = R. Then 2 2 2 j ∈ R. If j√≥ 0, then there exists a non-negative element r in R such that j − r = 0. Thus j = ± r and j ∈ R. This contradiction proves that j2 < 0. By Claim 4, j2 < 0. Thus there exists a positive element r ∈ R such that j2 = −r. √j 2 Replacing j by r , we can assume that j = −1. Take a non-zero element x in D−. Then the map D+ → D− defined by y 7→ yx is a + − C-linear map. Moreover it is a C-linear . Hence dimC D = dimC D = 1. Thus dim DR = 4. Let k = ij. Then j, k form a basis of D− over R and 1, i, j, k form a basis of D over R. Since j, k ∈ D−, they anticommute with i. This together with i2 = j2 = −1 and k = ij prove that 1, i, j, k satisfy the multiplication table given above for the H for R. ¤

References [1] Bochnak, J., Coste, M. and Roy, M. F., G´eom´etieAlg´ebriqueR´eelle, Springer Verlag, Berlin- Heidelberg-New York, (1987). [2] van den Dries, L., Tame and O-minimal Structure, London Math. Soc. Lect. Notes Ser. 248, Cambridge Univ. Press, Cambridge, (1998). [3] F. G. Frobenius, Uber¨ lineare Substitutionen und bilineare Formen, Journal f¨urdie reine und ange- wandte Mathematik 84 (1878), 1–63. [4] R. S. Palais, The Classification of real division algebras, Amer. Math. Monthly 75 (1968), 366–368.

Department of Mathematics, Faculty of Education, Wakayama University, Sakaedani Wakayama 640-8510, Japan, Partially supported by Kakenhi (23540101).

Department of Mathematics, Kyoto Prefectural University of Medicine, 13 Nishi- Takatsukaso-Cho, Taishogun Kita-ku, Kyoto 603-8334, Japan, Partially supported by Kakenhi (23540101) E-mail address: [email protected] E-mail address: [email protected]