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In a previous paper, we have given asymptotic formulas for the number of isomorphism
classes of -extensions with discriminant up to a given bound, both when the
signature of the extensions is or is not specified. We have also given very efficient
exact formulas for this number when the signature is not specified. The aim of this paper
is to give such exact formulas when the signature is specified. The problem is
complicated by the fact that the ray class characters which appear are not all genus
characters....
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