Galois groups in a family of dynatomic polynomials

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 ${\mathbb{Q}}_p$.