(~) by Springer-Verlag 1975 Abelian Extensions of Arbitrary Fields

Let k be an Hilbertian field, i.e. a field for which Hilbert's irreducibility theorem holds (cf. [1, 5]). It is obvious that the degree of the algebraic closure k of k is infinite with respect to k. It is not obvious that the same is true for the maximal p-extension of k, p a prime number. Let A be a finite abelian group. The questio