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Construction of tables of quartic number fields. (English) Zbl 0987.11079
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, 257-268 (2000).
The authors develop a new method for computing tables of quartic fields of small discriminant. Instead of using tools from the geometry of numbers, they evaluate a suitable Dirichlet series whose coefficients give the exact number of relative quadratic extensions of a field for any value of the norm of the relative discriminant. These computations can also be used to determine generating elements for the corresponding field extensions. While this is sufficient to construct tables of quartic fields of Galois group \(D_4\), the computations for \(A_4\) extensions \(L\) are more complicated. The Galois closure of these fields contains a cyclic cubic field \(K_3\) (coming from the cubic resolvent) and three quadratic extensions \(L_i\) of \(K_3\) from which the quartic \(A_4\) field \(L\) can be recovered. Similarly, but with even more effort, \(S_4\) extensions can be calculated, but this case is not treated in the present paper. The authors claim that they can produce tables of those fields which are a thousand times bigger than existing ones.
For the entire collection see [Zbl 0960.00039].
Reviewer: M.Pohst (Berlin)

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