Linearized Polynomials, Galois Groups and Symmetric Power Modules

TL;DR

This paper explores the Galois groups of linearized polynomials over function fields, demonstrating how symmetric powers of root modules can be realized within splitting fields of related polynomials.

Contribution

It introduces a novel realization of symmetric power modules of roots as subspaces in splitting fields, advancing understanding of Galois actions on linearized polynomials.

Findings

Galois groups of linearized polynomials over $\

$ ext{F}_q(t)$ are characterized.

Symmetric powers of root modules are embedded in splitting fields of related polynomials.

Abstract

We investigate some Galois groups of linearized polynomials over fields such as $\mathbb{F}_q(t)$. The space of roots of such a polynomial is a module for its Galois group. We present a realization of the symmetric powers of this module, as a subspace of the splitting field of another linearized polynomial.