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Generating functions for the numbers of abelian extensions of a local field

The aim of this paper is to give an explicit formula for the numbers of abelian extensions of a p-adic number field and to study the generating function of these numbers. More precisely, we give the number of abelian extensions with given degree and ramification index, and the number of abelian extensions with given degree of any local field of characteristic zero. Moreover, we give a concrete expression of a generating function for these last numbers.