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Cohen, Henri

Enumerating quartic dihedral extensions of \(\mathbb Q\) with signatures. (English) Zbl 1114.11085

Ann. Inst. Fourier 53, No. 2, 339-377 (2003).
Summary: In a previous paper with F. Diaz y Diaz and M. Olivier [Compos. Math. 133, No. 1, 65–93 (2002; Zbl 1050.11104)], we have given asymptotic formulas for the number of isomorphism classes of \(D_4\)-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.

Cited in 4 Documents

MSC:

11R16 Cubic and quartic extensions
11R29 Class numbers, class groups, discriminants
11R45 Density theorems
11Y40 Algebraic number theory computations

Keywords:

discriminant counting; genus character; quartic reciprocity

Citations:

Zbl 1050.11104
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\textit{H. Cohen}, Ann. Inst. Fourier 53, No. 2, 339--377 (2003; Zbl 1114.11085)
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