On fields of algebraic numbers with bounded local degrees

We provide a characterization of infinite algebraic Galois extensions of
the rationals with uniformly bounded local degrees.

In particular we show that for an infinite 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 an infinite extension
in some special cases.

We relate the uniform boundedness of the local degrees to
other properties: being a subfield of Q^(d), which is defined as the
compositum
of all number fields of degree at most d over Q; 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.