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

TL;DR

This paper characterizes the Galois group of certain linearized polynomials over fields of prime characteristic, showing it is the special linear group for prime degree polynomials under specific conditions.

Contribution

It proves that the Galois group of a monic q-polynomial of prime degree with specific coefficients is the special linear group, extending understanding of Galois groups in positive characteristic.

Findings

Galois group is SL(r,q) for prime degree r when q>2.

Galois group is PSL(r,q) in the projective case.

Analysis provided for q=2 case.

Abstract

Let $F$ be a field of prime characteristic $p$ and let $q$ be a power of $p$. We assume that $F$ contains the finite field of order $q$. A $q$-polynomial $L$ over $F$ is an element of the polynomial ring $F[x]$ with the property that those powers of $x$ that occur as terms of $L$ with nonzero coefficient have exponent a power of $q$. If the exponent of the leading term of $L$ is $q^n$, we say that $L$ has $q$-degree $n$. We assume that the coefficient of the $x$ term of $L$ is nonzero. We investigate the Galois group $G$, say, of $L$ over $F$, under the assumption that $L(x)/x$ is irreducible in $F[x]$. It is well known that if $L$ has $q$-degree $n$, its roots are an $n$-dimensional vector space over the field of order $q$ and $G$ acts linearly on this space. Our main theorem is the following. We consider a monic $q$-polynomial $L$ whose $q$-degree is a prime, $r$, say, with $L(x)/x$ irreducible over $F$. Assuming that the coefficient of the $x$ term of $L$ is $(-1)^r$, we show that the Galois group of $L$ over $F$ must be the special linear group $SL(r,q)$ of degree $r$ over the field of order $q$ when $q>2$. We also prove a projective version for the projective special linear group $PSL(r,q)$, and we present the analysis when $q=2$.