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Galois groups over rational function fields and explicit Hilbert irreducibility

David Krumm, Nicole Sutherland

TL;DR

Methods for computing the group G and obtaining an explicit description of the exceptional numbers c, i.e., those for which P(c,x)$ has Galois group different from $G$ or factors differently from $P$ are discussed.

Abstract

Let $P\in\mathbb Q[t,x]$ be a polynomial in two variables with rational coefficients, and let $G$ be the Galois group of $P$ over the field $\mathbb Q(t)$. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$ the specialized polynomial $P(c,x)$ has Galois group isomorphic to $G$ and factors in the same way as $P$. In this paper we discuss methods for computing the group $G$ and obtaining an explicit description of the exceptional numbers $c$, i.e., those for which $P(c,x)$ has Galois group different from $G$ or factors differently from $P$. To illustrate the methods we determine the exceptional specializations of three sample polynomials. In addition, we apply our techniques to prove a new result in arithmetic dynamics.

Galois groups over rational function fields and explicit Hilbert irreducibility

TL;DR

Methods for computing the group G and obtaining an explicit description of the exceptional numbers c, i.e., those for which P(c,x)GP$ are discussed.

Abstract

Let be a polynomial in two variables with rational coefficients, and let be the Galois group of over the field . It follows from Hilbert's Irreducibility Theorem that for most rational numbers the specialized polynomial has Galois group isomorphic to and factors in the same way as . In this paper we discuss methods for computing the group and obtaining an explicit description of the exceptional numbers , i.e., those for which has Galois group different from or factors differently from . To illustrate the methods we determine the exceptional specializations of three sample polynomials. In addition, we apply our techniques to prove a new result in arithmetic dynamics.

Paper Structure

This paper contains 15 sections, 22 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. An explicit form of HIT
  3. Resolvent polynomial interpretation
  4. Computation of Galois groups over $\mathbb{Q}(t)$
  5. Galois groups of irreducible polynomials over $\mathbb{Q}(t)$
  6. Galois groups of reducible polynomials over $\mathbb{Q}(t)$
  7. Computing a fixed field of a subgroup of a Galois group
  8. Proof of Galois groups of polynomials over $\mathbb{Q}(t)$
  9. Examples
  10. A finite exceptional set
  11. An infinite family of exceptional factorizations
  12. An infinite family of exceptional Galois groups
  13. A reducible polynomial
  14. A polynomial of degree 30
  15. An application to arithmetic dynamics

Key Result

Theorem 1

Let $\Delta(t)$ and $\ell(t)$ be the discriminant and leading coefficient of $P$, respectively. Let $M_1,\ldots, M_r$ be representatives of all the conjugacy classes of maximal subgroups of $G$. For $i=1,\ldots, r$, let $F_i$ be the fixed field of $M_i$ and let $f_i(t,x)$ be a monic irreducible poly Then $c\in\mathcal{E}(P)$$\iff$ there is an index $i$ such that $f_i(c,x)$ has a root in $\mathbb{Q

Theorems & Definitions (43)

  • Theorem : see \ref{['main_hit_thm']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 33 more