Algebra & Number Theory
- Algebra Number Theory
- Volume 13, Number 4 (2019), 963-993.
A finiteness theorem for specializations of dynatomic polynomials
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 .
Algebra Number Theory, Volume 13, Number 4 (2019), 963-993.
Received: 28 May 2018
Revised: 22 January 2019
Accepted: 22 February 2019
First available in Project Euclid: 18 May 2019
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Krumm, David. A finiteness theorem for specializations of dynatomic polynomials. Algebra Number Theory 13 (2019), no. 4, 963--993. doi:10.2140/ant.2019.13.963. https://projecteuclid.org/euclid.ant/1558144826