Exact computation of the discriminants of Abelian extensions. (Comptage exact de discriminants d’extensions abéliennes.) (French) Zbl 0976.11055

Cohen, Henri

J. Théor. Nombres Bordx. 12, No. 2, 379-397 (2000).

The author gives an exact count of the number of automorphism classes of abelian extensions of degree \(\geq 4\) by using the Dirichlet series for enumerating such fields (from class field theory). The secret lies in powerful summatory methods based on the classical “hyperbola” method for summing the divisor function. An error estimate is also given for the computer time.

Reviewer: Harvey Cohn (San Diego)

MSC:

11Y40	Algebraic number theory computations
11R20	Other abelian and metabelian extensions
11R29	Class numbers, class groups, discriminants

Keywords:

discriminant density; automorphism classes; abelian extensions; Dirichlet series; summatory methods; error estimate; computer time

References:

[1]	Cohen, H., A course in computational algebraic number theory (third printing). , Springer-Verlag (1996). · Zbl 0786.11071
[2]	Cohen, H., Advanced topics in computational number theory. , Springer-Verlag (2000). · Zbl 0977.11056
[3]	Cohen, H., Diaz Y Diaz, F., Olivier, M., Densité des discriminants des extensions cycliques de degré premier, C.R. Acad. Sci. Paris330 (2000), 61-66. · Zbl 0941.11042
[4]	Cohen, H., Diaz Y Diaz, F., Olivier, M., Counting discriminants of number fields of degree up to four. proceedings ANTS IVLeiden (2000), , Springer-Verlag, 269-283. · Zbl 0987.11080
[5]	Cohen, H., Diaz Y Diaz, F., Olivier, M., Counting discriminants of number fields. MSRI preprint2000-026 (2000), 9p. · Zbl 0987.11080
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