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Enumerating quartic dihedral extensions of $${\mathbb Q}$$. (English) Zbl 1050.11104
The main result of this paper is an explicit formula for the number of isomorphism classes of $$D_4$$ extensions of $$\mathbb{Q}$$, ie. non-normal number fields of degree $$4$$ where the Galois group of the normal closure is a dihedral group with $$8$$ elements. Assuming additional conjectures on the density of $$S_4$$ extensions it is shown that the $$D_4$$ fields have a positive density among all degree $$4$$ fields. The (explicit) formulas are based on the analytical continuation of the associated Dirichlet series which allows the calculation of the residue at the simple pole $$s=1$$.
The result is based on two key ingredients: the first is an (explicit) parametrization of quadratic extensions of any number field based on the notion of $$2$$-Selmer group and “virtual units” of the base field. The classification depends on the knowledge of the $$2$$-part of the ideal class group and the unit group of the base field. The second ingredient is classical genus theory for quadratic fields.
To enumerate $$D_4$$ extension, the authors consider all relative quadratic extensions of quadratic fields. An easy argument allows them to seperate the normal fields (fields with Galois group isomorphic to $$C_4$$ or $$V_4$$). By careful application of genus theory they are able to control the $$2$$ parts of the class groups of the intermediate fields and thus obtain explicit formulas for the number of degree $$4$$ extensions of this shape as well as for the asymptotic behaviour of the counting function.
The power of their method over pure class field theoretical methods like C. Fieker and J. Klüners [J. Number Theory 99, No. 2, 318–337 (2003; Zbl 1049.11139)] is demontrated by the tables in the end where the numbers of such fields for discriminants up to $$10^{17}$$ are given. Furthermore, the authors explain how to obtain the fields themselves rather than the number only.

##### MSC:
 11Y40 Algebraic number theory computations 11R16 Cubic and quartic extensions 11R29 Class numbers, class groups, discriminants 11R45 Density theorems 11R11 Quadratic extensions
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