  Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be an endomorphism of degree $d\ge2$
defined over $\overline{\mathbb{Q}}$ or $\overline{\mathbb{Q}}_p$, and let $K$
be the field of moduli of $f$. We prove that there is a field of definition $L$
for $f$ whose degree $[L:K]$ is bounded solely in terms of $N$ and $d$.
