Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert’s irreducibility theorem and a -adic form à la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over . The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.
This paper considers some refined versions of the Inverse Galois Problem. We study the local or global behavior of rational specializations of some finite Galois covers of .
This article examines the realizability of groups of order 64 as Galois groups over arbitrary fields. Specifically, we provide necessary and sufficient conditions for the realizability of 134 of the 200 noncyclic groups of order 64 that are not direct products of smaller groups.
Let p be an odd prime and k an arbitrary field of characteristic not p. We determine the obstructions for the realizability as Galois groups over k of all groups of orders p 5 and p 6 that have an abelian quotient obtained by factoring out central subgroups of order p or p 2. These obstructions are decomposed as products of p-cyclic algebras, provided that k contains certain roots of unity.
2000 Mathematics Subject Classification: 12F12.We find the obstructions to realizability of groups of order 32 as Galois groups over arbitrary field of characteristic not 2. We discuss explicit extensions and automatic realizations as well.This work is partially supported by project of Shumen University