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A note on Galois groups of linearized polynomials

Peter Müller

Abstract

Let $L(X)$ be a monic $q$-linearized polynomial over $F_q$ of degree $q^n$, where $n$ is an odd prime. Recently Gow and McGuire showed that the Galois group of $L(X)/X-t$ over the field of rational functions $F_q(t)$ is $GL_n(q)$ unless $L(X)=X^{q^n}$. The case of even $q$ remained open, but it was conjectured that the result holds too and partial results were given. In this note we settle this conjecture. In fact we use Hensel's Lemma to give a unified proof for all prime powers $q$.

A note on Galois groups of linearized polynomials

Abstract

Let be a monic -linearized polynomial over of degree , where is an odd prime. Recently Gow and McGuire showed that the Galois group of over the field of rational functions is unless . The case of even remained open, but it was conjectured that the result holds too and partial results were given. In this note we settle this conjecture. In fact we use Hensel's Lemma to give a unified proof for all prime powers .
Paper Structure (2 sections, 4 theorems, 2 equations)

This paper contains 2 sections, 4 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Proof of Proposition \ref{['p:main']}

Key Result

Theorem 1

Let $q$ be a prime power and $n$ be an odd prime. Let $L(X)$ be a monic $q$-linearized polynomial over $\mathbb F_q$ of $q$-degree $n$. Then the Galois group of $L(X)/X-t$ over $\mathbb F_q(t)$ is $\mathop{\mathrm{GL}}\nolimits_n(q)$ unless $L(X)=X^{q^n}$.

Theorems & Definitions (8)

  • Theorem 1
  • Proposition 2
  • Lemma 3
  • proof
  • Remark 4
  • Corollary 5
  • proof
  • Remark 6