Constant rank subspaces of alternating bilinear forms from Galois Theory

TL;DR

This paper investigates the structure of alternating bilinear forms over cyclic Galois extensions, showing how these forms can be decomposed into subspaces with constant rank properties, especially under certain algebraic conditions.

Contribution

It extends known results by demonstrating that components of these form spaces can decompose into sums of constant-rank subspaces, providing new insights into their structure.

Findings

Decomposition of $A^\sigma$ into sums of constant rank subspaces.

Existence of a subspace of dimension $n/2$ with all nonzero forms non-degenerate.

Constant-rank decompositions in several algebraic situations.

Abstract

Let $L/K$ be a cyclic extension of degree $n = 2m$. It is known that the space $\text{Alt}_K(L)$ of alternating $K$-bilinear forms (skew-forms) on $L$ decomposes into a direct sum of $K$-subspaces $A^{\sigma^i}$ indexed by the elements of $\text{Gal}(L/K) = \langle \sigma \rangle$. It is also known that the components $A^{\sigma^i}$ can have nice constant-rank properties. We enhance and enrich these constant-rank results and show that the component $A^\sigma$ often decomposes directly into a sum of constant rank subspaces, that is, subspaces all of whose non-zero skew-forms have a fixed rank $r$. In particular, this is always true when $-1 \not \in L^2$. As a result we deduce a decomposition of $\text{Alt}_K(L)$ into subspaces of constant rank in several interesting situations. We also establish that a subspace of dimension $\frac{n}{2}$ all of whose nonzero skew-forms are non-degenerate can always be found in $A^{\sigma^i}$ where $\sigma^i$ has order divisible by $2$.