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

2022/07/28 by Rod Gow, G. E. McGuire, Gow, Rod +1 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2207.14113

openalex publication_date 2022/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Cited by

• Effective Artin-Schreier-Witt theory for curves

Related

• On Galois groups of linearized polynomials related to the special linear group of prime degree
• Linearized Polynomials, Galois Groups and Symmetric Power Modules
• Realizing cyclic linear transformations as Frobenius elements in the Galois groups of q-polynomials over function fields
• Linearization of polynomials in prime characteristic, with applications to the Golay code and Steiner system
• On the number of roots of some linearized polynomials