Méthodes géométriques dans la recherche des petits discriminants. (French) Zbl 0567.12009

Théorie des nombres, Sémin. Delange-Pisot-Poitou, Paris 1983-84, Prog. Math. 59, 147-179 (1985).
[For the entire collection see Zbl 0561.00004.]
This paper gives a survey of the methods for finding, for given \(r,s\), and \(M\), the fields with discriminants less than \(M\) in absolute value, having \(r\) real and \(s\) pairs of complex conjugates. Methods from the geometry of numbers yield a number of inequalities for the coefficients of a defining polynomial of such a field. If \(n=r+2s\) is composite the possibility that a polynomial may merely define a subfield is considered, as in the question of deciding whether two fields with the same discriminant are isomorphic. The extensive bibliography is supplemented by notes that give a lot of details about the results that have so far been obtained, mainly for \(n\leq 8\).
Reviewer: H.J.Godwin


11R29 Class numbers, class groups, discriminants
11R21 Other number fields
11R16 Cubic and quartic extensions


Zbl 0561.00004