Abstract
The Beckmann-Black problem asks whether any given finite Galois extension E/K of group G is the specialization at some point of some finite regular Galois extension with the same group. In this paper, we study a generalization of this problem for infinite extensions, via the profinite twisting lemma, and apply the latter in many situations, like abelian infinite extensions and the -universal Frattini cover of an arbitrary finite group.
KEYWORDS:
2020 MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgments
I wish to thank P. Dèbes for very helpful comments and many valuable suggestions. Also, I am indebted to an anonymous reviewer for insightful comments and a kind help and his interest in my paper.
Notes
1 Unramified point in the extension here means that the ideal is unramified in
2 A field K is said to be an ample field if each geometrically irreducible smooth curve defined over K with at least one K-rational point has infinitely many such points (for more detail, see [3, 8, 14, 18]).
3 This means that t is globally invariant under the action of GK and that each coordinate is separable.
4 A field K is said to be a real closed field if K is an ordered field for which no non-trivial algebraic extension can be ordered. The field of real numbers is a real closed field.
5 The viewpoints of cover and field extensions are equivalent, thus allowing to define a regular realization in terms of cover.
6 The branch point set of a cover is the finite set of points such that the associated discrete valuations are ramified in the corresponding function field extension
7 We can drop the hypothesis “K is an uncountable field” in proportion 3.1 of [13] because we assume that the tower of G-extensions has an unramified point
8 In similar proof as in Corollary 4.2, we have the same result for K is real closed field. e.g. because it is a regular ψ-free field (see [9, theorem 2.3]) and for is a field of formal Laurent series with coefficients in because it is a regular ψ-free field (see [9, Section 3.2]).