Fields of algebraic numbers with bounded local degrees and their properties

TL;DR

This paper characterizes infinite Galois extensions of the rationals with bounded local degrees, linking these properties to the structure of their Galois groups and providing explicit bounds and counterexamples.

Contribution

It establishes the equivalence of bounded local degrees and finite exponent Galois groups, generalizes previous results, and constructs explicit counterexamples using group theory.

Findings

Bounded local degrees are equivalent to Galois groups of finite exponent.

Explicit formulas for bounds on local degrees in special cases.

Counterexamples show the properties are not always equivalent in non-abelian cases.

Abstract

We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees, giving a detailed proof of all the results announced in a paper by Checcoli and Zannier and obtaining relevant generalizations for them. In particular we show that that for an infinite Galois extension of the rationals the following three properties are equivalent: having uniformly bounded local degrees at every prime; having uniformly bounded local degrees at almost every prime; having Galois group of finite exponent. The proof of this result enlightens interesting connections with Zelmanov's work on the Restricted Burnside Problem. We give a formula to explicitly compute bounds for the local degrees of a infinite extension in some special cases. We relate the uniform boundedness of the local degrees to other properties: being a subfield of the compositum of all number fields of bounded degree; being generated by elements of bounded degree. We prove that the above properties are equivalent for abelian extensions, but not in general; we provide counterexamples based on group-theoretical constructions with extraspecial groups and their modules, for which we give explicit realizations.