Fe b 20 07 Prime to p extensions of the generic abelian crossed product

In this paper we prove that the noncyclic generic abelian crossed product p-algebras constructed by Amitsur and Saltman in [AS78] remain noncyclic after tensoring by any prime to p extension of their centers. We also prove that an example due to Saltman of an indecomposable generic abelian crossed product with exponent p and degree p 2 remains indecomposable after any prime to p extension. 0 Introduction Let p be a prime. A p-algebra is a finite dimensional division algebra which is central over a field of characteristic p and has index a power of p. The question of whether or not every p-algebra is cyclic was first asked by Albert in [Alb61]. It was a natural question to ask as he had proven in [Alb39] that every p-algebra is similar in the Brauer group of its center to a cyclic algebra. Amitsur and Saltman answered the question in [AS78] by constructing, for any fixed prime p, generic abelian crossed product p-algebras which contain no p-power central elements, that is, non-central elements whose p-th power is central. This proved the existence of noncyclic p-algebras of all degrees p n , n ≥ 2. Credit for this problem is also due to Isaac Gordon who constructed a noncyclic 2-algebra of degree 4 in 1940. Let D be a finite dimension division algebra with center F and let E/F be an extension of fields. We call D E = D ⊗ F E a prime to p extension of D if the degree of E/F is prime to p. The main results of this paper, Theorem 3.9 and Corollary 3.10, prove that there do not exist prime to p extensions of the Amitsur-Saltman noncyclic generic abelian crossed products which are cyclic. This is in the opposite direction of the well known result that every division algebra of prime degree does become cyclic after a prime to p extension. Other results regarding prime to p extensions of p-power index division algebras can be found in [RS92]. The outline of this paper is as follows. In section 1 we determine precise conditions under which abelian crossed product algebras have homogeneous p-power central elements. This condition, which is placed on the matrix defining the algebra, is called