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

##### MSC:
 11Y40 Algebraic number theory computations 11R20 Other abelian and metabelian extensions 11R29 Class numbers, class groups, discriminants
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##### 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 [6] Cohen, H., Diaz Y Diaz, F., Olivier, M., Asymptotic and exact enumemtion of discriminants of number fields. En préparation. [7] Cohen, H., Diaz Y Diaz, F., Olivier, M., Enumerating quartic dihedral extensions of Q. Submitted. · Zbl 1050.11104 [8] Cohen, H., Diaz Y Diaz, F., Olivier, M., On the density of discriminants of cyclic extensions of prime degree. En préparation. · Zbl 1004.11063 [9] Cohen, H., Diaz Y Diaz, F., Olivier, M., On the density of discriminants of quartic number fields. En préparation. · Zbl 1193.11109 [10] Cohen, H., Dress, F., El Marraki, M., Explicit estimates for summatory functions linked to the Möbius μ- function. Preprint (1996), soumis. · Zbl 1230.11118 [11] Deléglise, M., Rivat, J., Computing the summation of the Möbius function. Exp. Math.5 (1996), 291-295. · Zbl 1007.11083 [12] Tenenbaum, G., Introduction à la théorie analytique et probabiliste des nombres. Cours Spécialisé S.M.F. 1, Paris (1996). · Zbl 0880.11001
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