A CONDITION FOR CYCLIC ALGEBRAS TO BE SPLIT

Abstract

(cf. for instance [1, 30.6, p. 415]). This criterion translates the question of whether A is split into the problem whether or not a certain element is represented by a homogeneous form (the reduced norm of l/k) of degree d in d indeterminates. In order to give a sufficient criterion for A to be split it is sometimes enough, however, to check a much simpler?binary?form of degree d that arises by restricting the reduced norm of the algebra A to a suitable subspace of A: let & be a field of characteristic not 3 that contains a primitive third root of unity, and let A = (I, a) be a cyclic algebra over k with / = k[x]/(x3 ? b) cubic etale. If the binary cubic form (a, b) represents 1 or 62, or if the binary cubic form (a, b2) represents 1 or 6, then A splits, i.e. A = Mat3(A:). More generally, take any odd integer d and let k be a field of characteristic not dividing d containing a primitive dth root of unity. Suppose that A = (/, a) is a cyclic central simple algebra over k of degree d with / = k[x]/(xd ? b), a field extension of k. Then A splits if there is an integer r, 1 < r < d ? 1, such that the form (a, br) of degree d represents bs for some integer s with r^s, 0<s<d ? 1. Also for certain 27-dimensional exceptional simple cubic Jordan algebras it suffices to check a much simpler 6-dimensional cubic form to see if the algebra