Linear Algebra and Galois Theory

TL;DR

This paper explores the application of Galois theory to linear algebra, providing new insights into endomorphism structures, trace, and linear independence through tensor algebra methods.

Contribution

It offers a Galois-theoretic perspective on linear algebraic objects, with shorter, more conceptual proofs using tensor algebra.

Findings

Clarification of rank-one endomorphism structure

Criteria for linear independence derived from Galois theory

Simplified proofs using tensor algebra

Abstract

In \cite{GQ2008} R. Gow and R. Quinlan have cast a new look on the endomorphism algebra of a $K$-vector space $V$ of dimension $n$ assuming that $K$ has a Galois extension $L$ of degree $n$. In this approach the $K$-space $L$ may serve as a model for $V$ and Galois-theoretic ideas and results may be applied to elucidate the structure of endomorphisms and other important objects of linear algebra. In particular, this leads to the clarification of the structure of a rank-one endomorphism, trace of an endomorphism, criteria for linear indepedence etc. We present an exposition of these results using the language of tensor algebra wherever possible to provide shorter and more conceptual proofs.