Generic Splitting Fields of Composition Algebras
1967; American Mathematical Society; Volume: 128; Issue: 3 Linguagem: Inglês
10.2307/1994471
ISSN1088-6850
Autores Tópico(s)Rings, Modules, and Algebras
ResumoWitt [7] proved that one can assign to each generalized quaternion algebra si over a field K, a field F(si) containing K which splits si and has the property: if F(si) splits a quaternion algebra Se over K then either á? is split over K or 3 § is isomorphic to si.Amitsur [2] has generalized this result to obtain generic splitting fields for all central simple associative algebras of dimension greater than one over K (cf.Roquette [6]).In this paper we generalize the result of Witt in another direction, studying splitting fields of composition algebras of dimension greater than one over K of characteristic other than two.We assign to each such algebra (€, a field F($>) containing K, prove that F(<^) is an invariant under isomorphisms, and prove Theorem 2. Let ^ be a composition algebra of dimension greater than one over K.Then 1. #*■(«■) is split.2. If F^K is any field, then ^F is split if and only if there is a K-place of FÇ£)into F >J oo.3. IfW is any composition algebra over K such that Wp^ is split, then either c€' is split or *€ is isomorphic to a subalgebra of'S'.Thus we generalize the result of Witt to quadratic and generalized Cayley algebras.I. Composition algebras.A composition algebra # over a field K is an algebra over K, with identity 1, together with a nondegenerate quadratic form TV such that N(xy) = N(x)N(y) for any x, y in (&.The structure of such algebras has been completely determined and we refer to [1] or [4] for proofs of the following results.1.A composition algebra <€ is alternative with involution t:o.I+u^-al-u, for u orthogonal to 1 with respect to the nondegenerate, symmetric, bilinear form N(x,y)=^{N(x+y) -N(x)-N(y)}.Each xetf can be uniquely represented in the form x = al + u, aeK, N(u, 1) = 0 and one has TV(x)l =(al + M)(al-«).If F is a subspace of (€, we shall denote by V1 the orthogonal complement of F in # with respect to TV(x, y).2. If âS is a composition subalgebra of <€ (necessarily having associated quadratic form the restriction of TV to 3S), and ke^sC, TV(w)5¿0, then 3S + 3Su,
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