Minimum Ramification for Finite Abelian Extensions Over Q and Q(i)

Abstract

The minimum number of finite primes that must ramify for a given finite abelian group to be realized over Q and Q(i) is given. To this end, we give a criterion for the existence of a surjection from a finitely generated abelian pro-pi group onto a finite abelian group. Class field theory is then applied to find the Galois group of the maximal abelian extension over Q and Q(i). The surjection criterion is then applied to each case to solve the minimum ramification of each finite abelian group.