Cycle-lengths of a class of monic binomials

Abstract

Let $K$ be an algebraic field of degree $N$ and let $p$ be an odd prime. It is shown that if $K$ does not contain $p$-th primitive roots of unity and $f(X)=X^{p^k}+c$ with $k\ge1$ and non-zero $c\in K$, then the length of cycles of $f$ in $K$ is bounded by a value depending only on $K$ and $p$. If $p>2^N$, then this bound depends only on $N$.