Galois groups in a family of dynatomic polynomials

TL;DR

This paper analyzes the Galois groups of the fourth dynatomic polynomial for quadratic polynomials over rationals, revealing their structure and implications for a conjecture on periodic points in local fields.

Contribution

It determines the Galois group structure and factorization degrees of $\

Findings

Identifies the Galois group structure for $\

Shows that over 39% of primes, quadratic polynomials lack points of period four in $\

Provides new insights related to the Morton-Silverman boundedness conjecture.

Abstract

For every nonconstant polynomial $f\in\mathbb Q[x]$, let $\Phi_{4,f}$ denote the fourth dynatomic polynomial of $f$. We determine here the structure of the Galois group and the degrees of the irreducible factors of $\Phi_{4,f}$ for every quadratic polynomial $f$. As an application we prove new results related to a uniform boundedness conjecture of Morton and Silverman. In particular we show that if $f$ is a quadratic polynomial, then, for more than $39\%$ of all primes $p$, $f$ does not have a point of period four in $\mathbb Q_p$.