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Enumerating Quartic Dihedral Extensions of $\open Q$

Published online by Cambridge University Press:  04 December 2007

Henri Cohen ,
Francisco Diaz Y Diaz  and
Michel Olivier
Henri Cohen
Affiliation:
Laboratoire A2X, UMR 5465 du CNRS, Université Bordeaux I, 351 Cours de la Libération, 33405 TALENCE Cedex, France. E-mail: cohen@math.u-bordeaux.fr
Francisco Diaz Y Diaz
Affiliation:
Laboratoire A2X, UMR 5465 du CNRS, Université Bordeaux I, 351 Cours de la Libération, 33405 TALENCE Cedex, France. E-mail: diaz@math.u-bordeaux.fr
Michel Olivier
Affiliation:
Laboratoire A2X, UMR 5465 du CNRS, Université Bordeaux I, 351 Cours de la Libération, 33405 TALENCE Cedex, France. E-mail: olivier@math.u-bordeaux.fr
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Abstract

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We give an explicit Dirichlet series for the generating function of the discriminants of quartic dihedral extensions of $\open Q$. From this series we deduce an asymptotic formula for the number of isomorphism classes of such quartic extensions with discriminant up to a given bound. On the other hand, by using essentially classical results of genus theory combined with elementary analytical methods such as the method of the hyperbola, we show how to compute exactly this number up to quite large bounds, and we give a table of selected values.

Keywords

discriminant densitiesD4-extensions
Type
Research Article
Information
Compositio Mathematica , Volume 133 , Issue 1 , August 2002 , pp. 65 - 93
Copyright
© 2002 Kluwer Academic Publishers