Let and be indeterminates, let , and for every positive integer let denote the -th dynatomic polynomial of . Let be the Galois group of over the function field , and for let be the Galois group of the specialized polynomial . It follows from Hilbert’s irreducibility theorem that for fixed we have for every outside a thin set . By earlier work of Morton (for ) and the present author (for ), it is known that is infinite if . In contrast, we show here that is finite if . As an application of this result we show that, for these values of , the following holds with at most finitely many exceptions: for every , more than of prime numbers have the property that the polynomial does not have a point of period in the -adic field .
"A finiteness theorem for specializations of dynatomic polynomials." Algebra Number Theory 13 (4) 963 - 993, 2019. https://doi.org/10.2140/ant.2019.13.963