Parametric Galois Extensions

Given a field $k$ and a finite group $H$, {\it{an $H$-parametric extension over $k$}} is a finite Galois extension of $k(T)$ of Galois group containing $H$ which is regular over $k$ and has all the Galois extensions of $k$ of group $H$ among its specializations. We are mainly interested in producing non $H$-parametric extensions, which relates to classical questions in inverse Galois theory like the Beckmann-Black problem and the existence of one parameter generic polynomials. We develop a general approach started in a preceding paper and provide new non parametricity criteria and new examples.