Galois extensions and subspaces of alternating bilinear forms with special rank properties

Abstract

Let K be a field admitting a cyclic Galois extension of degree n . The main result of this paper is a decomposition theorem for the space of alternating bilinear forms defined on a vector space of odd dimension n over K . We show that this space of forms is the direct sum of ( n - 1 ) / 2 subspaces, each of dimension n , and the non-zero elements in each subspace have constant rank defined in terms of the orders of the Galois automorphisms. Furthermore, if ordered correctly, for each integer k lying between 1 and ( n - 1 ) / 2 , the rank of any non-zero element in the sum of the first k subspaces is at most n - 2 k + 1 . Slightly less sharp similar results hold for cyclic extensions of even degree.