Minimal and characteristic polynomials of symmetric matrices in characteristic two

TL;DR

This paper characterizes which polynomials are minimal or characteristic polynomials of symmetric matrices over fields of characteristic two, linking algebraic properties to matrix realizations.

Contribution

It provides a complete characterization of minimal and characteristic polynomials of symmetric matrices in characteristic two, including conditions involving inseparable irreducible polynomials.

Findings

A polynomial is the minimal/characteristic polynomial of a symmetric matrix iff it is not a product of distinct inseparable irreducible polynomials.

Any algebraic element of degree n can be an eigenvalue of a symmetric matrix of size n or n+1.

The paper establishes the size bounds for symmetric matrices realizing given eigenvalues.

Abstract

Let $k$ be a field of characteristic two. We prove that a non constant monic polynomial $f\in k[X]$ of degree $n$ is the minimal/characteristic polynomial of a symmetric matrix with entries in $k$ if and only if it is not the product of pairwise distinct inseparable irreducible polynomials. In this case, we prove that $f$ is the minimal polynomial of a symmetric matrix of size $n$. We also prove that any element $\alpha\in k_{alg}$ of degree $n\geq 1$ is the eigenvalue of a symmetrix matrix of size $n$ or $n+1$, the first case happening if and only if the minimal polynomial of $\alpha$ is not the product of pairwise distinct inseparable irreducible polynomials.