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arXiv · 0709.0741

Galois extensions and subspaces of bilinear forms with special rank properties

Rod Gow·Rachel Quinlan
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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-2k+1. Slightly less sharp similar results hold for cyclic extensions of even degree.

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Rod Gow, Rachel Quinlan. 2007-09-05. Galois extensions and subspaces of bilinear forms with special rank properties. https://arxiv.org/abs/0709.0741

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