$$ We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli $ℝ$ that can not be defined over $ℝ$ is also given.

We continue the examination of the stable reduction and fields of moduli of $G$-Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $(0,p)$, where $G$ has a cyclic$p$-Sylow subgroup $P$ of order $p^n$. Suppose further that the normalizer of $P$ acts on $P$ via an involution. Under mild assumptions, if $f:Y\to {\mathbb{P}}^1$ is a three-point $G$-Galois cover defined over $\overline{\mathbb{Q}}$, then the $n$th higher ramification groups above $p$ for the upper numbering of the (Galois closure of the) extension $K/\mathbb{Q}$ vanish,...

Let $k$ be a number field, ${𝒪}_k$ its ring of integers, and $f(t,X)\in k\left(t\right)\left[X\right]$ be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations $t\mapsto \overline{t}\in {𝒪}_k$ such that $f(\overline{t},X)$ is still irreducible. In this paper we study the set ${\mathrm{Red}}_f\left({𝒪}_k\right)$ of those $\overline{t}\in {𝒪}_k$ with $f(\overline{t},X)$ reducible. We show that ${\mathrm{Red}}_f\left({𝒪}_k\right)$ is a finite set under rather weak assumptions. In particular, previous results obtained by diophantine approximation techniques, appear as special cases of some of our results. Our method is different. We use elementary group...