Galois extensions and subspaces of bilinear forms with special rank properties

TL;DR

This paper presents a decomposition theorem for the space of alternating bilinear forms over fields with cyclic Galois extensions, revealing structured subspaces with constant rank properties related to Galois automorphisms.

Contribution

It introduces a novel decomposition of bilinear form spaces over fields with cyclic Galois extensions, linking subspace ranks to Galois automorphism orders.

Findings

Decomposition of alternating bilinear forms into direct sums of subspaces.

Constant rank properties within each subspace based on Galois automorphisms.

Bounds on ranks of sums of initial subspaces.

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.