On Galois groups of linearized polynomials related to the general linear group of prime degree

TL;DR

This paper investigates the Galois groups of certain linearized polynomials over finite fields, establishing that for prime degrees, the Galois group is typically the general linear group, with a specific exception.

Contribution

It proves that for prime degree, the Galois group of the polynomial is always GL(n,q), except for a specific monomial case, advancing understanding of polynomial Galois groups over finite fields.

Findings

Galois group is GL(n,q) for prime n

Exception occurs when L(x)=x^{q^n}

Results connect Galois groups to monodromy groups

Abstract

Let $L(x)$ be any $q$-linearized polynomial with coefficients in $\mathbb{F}_q$, of degree $q^n$. We consider the Galois group of $L(x)+tx$ over $\mathbb{F}_q(t)$, where $t$ is transcendental over $\mathbb{F}_q$. We prove that when $n$ is a prime, the Galois group is always $GL(n,q)$, except when $L(x)=x^{q^n}$. Equivalently, we prove that the arithmetic monodromy group of $L(x)/x$ is $GL(n,q)$.