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 \(\mathbb Q_p\).