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Realizing cyclic linear transformations as Frobenius elements in the Galois groups of $q$-polynomials over function fields

Rod Gow, Gary McGuire

Abstract

We realize Frobenius conjugacy classes in Galois groups of certain $q$-polynomials over $\mathbb{F}_q(t)$ using specific degree 1 ideals. We combine this with methods from elementary linear algebra and group theory to realize transvections in some linear Galois groups. This enables the Galois group to be identified as a known classical group in several reasonably general cases.

Realizing cyclic linear transformations as Frobenius elements in the Galois groups of $q$-polynomials over function fields

Abstract

We realize Frobenius conjugacy classes in Galois groups of certain -polynomials over using specific degree 1 ideals. We combine this with methods from elementary linear algebra and group theory to realize transvections in some linear Galois groups. This enables the Galois group to be identified as a known classical group in several reasonably general cases.
Paper Structure (14 sections, 41 theorems, 123 equations)

This paper contains 14 sections, 41 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Galois theory relating to ring extensions
  4. Some results from linear algebra
  5. Factorization of polynomials over function fields
  6. Construction of self-reciprocal polynomials
  7. The symplectic group
  8. Structure of the symplectic group
  9. Realizing symplectic transvections
  10. The orthogonal group of even dimension
  11. On $q$-anti-palindromic polynomials
  12. A nondegenerate quadratic form
  13. The orthogonal group as a Galois group
  14. A connection with the Weyl group

Key Result

Lemma 1

Let $L$ be a $q$-polynomial of $q$-degree $n$ in $\mathop{\mathrm{\mathbb{F}}}\nolimits_q(t)[x]$, with Let $E$ be a splitting field for $L$ over $\mathop{\mathrm{\mathbb{F}}}\nolimits_q(t)$ and let $V$ be the set of roots of $L$ in $E$. Let $G$ be the Galois group of $E$ over $F$. Suppose that $a_0(t)\neq 0$. Then $V$ is an $\mathop{\mathrm{\mathbb{F}}}\nolimits_q$-vector space of dimension $n$ a

Theorems & Definitions (67)

  • Lemma 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • proof
  • Corollary 5
  • Theorem 6
  • proof
  • Corollary 7
  • Lemma 8
  • ...and 57 more