HILBERT IRREDUCIBILITY, THE MALLE CONJECTURE AND THE GRUNWALD PROBLEM

Théorème d'irréductibilité de Hilbert, conjecture de Malle et problème de Grunwald

François Emmanuel Motte (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

The central result is a new explicit version of the Hilbert Irreducibility Theorem. Then, starting from a regular Galois extension F/K(T), we can count the number of specialized extensions F_{t_0}/K and not only the specialization points t_0, and provide some control of N_{K/Q}(d_{F_{t_0}}) Consequently, we contribute to the Malle conjecture on the number N(K,G,y) of finite Galois extensions E of some number field K of finite group G and of ideal discriminant of norm N_{K/Q}(d_E)< y. For every number field K containing a certain number field K_0 (depending on G), we establish this lower bound : N(K,G,y)

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

François Emmanuel Motte. HILBERT IRREDUCIBILITY, THE MALLE CONJECTURE AND THE GRUNWALD PROBLEM. 2019. ⟨hal-01965547v6⟩