On Galois groups of linearized polynomials related to the general linear group of prime degree
Rod Gow, Gary McGuire

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 be any -linearized polynomial with coefficients in , of degree . We consider the Galois group of over , where is transcendental over . We prove that when is a prime, the Galois group is always , except when . Equivalently, we prove that the arithmetic monodromy group of is .
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Coding theory and cryptography
