On the Local Behavior of Specializations of Function Field Extensions

Given a field k of characteristic zero and an indeterminate T over k, we investigate the local behavior at primes of k of finite Galois extensions of k arising as specializations of finite Galois extensions E/k(T) (with E/k regular) at points t₀ ϵ P¹(k). We provide a general result about decomposition groups at primes of k in specializations, extending a fundamental result of Beckmann concerning inertia groups. We then apply our result to study crossed products, the Hilbert-Grunwald property, and finite parametric sets.