Counting discriminants of number fields of degree up to four. (English) Zbl 0987.11080

Bosma, Wieb (ed.), Algorithmic number theory. 4th international symposium. ANTS-IV, Leiden, the Netherlands, July 2-7, 2000. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1838, 269-283 (2000).
For each possible Galois group of number fields of degree at most four, the authors determine the number of such fields with discriminant below a given bound. The method is based on the evaluation of the coefficients of suitable Dirichlet series. From that series, the authors deduce an asymptotic formula that is much easier to calculate. In each case they also provide numerical results for bounds up to \(10^{19}\). These show a remarkable correspondence between the exact number of fields and those predicted by the asymptotic formula.
For the entire collection see [Zbl 0960.00039].
Reviewer: M.Pohst (Berlin)


11Y40 Algebraic number theory computations
11R16 Cubic and quartic extensions
11R11 Quadratic extensions
11-04 Software, source code, etc. for problems pertaining to number theory