Galois extensions and subspaces of bilinear forms with special rank properties

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

2007-09-05

arxiv.org/abs/0709.0741v1

Mathematics

Commutative Algebra

13 pages

Scientific paper

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-2k+1. Slightly less sharp similar results hold for cyclic extensions of even degree.

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