Geometric methods in the study of small discriminants. (Méthodes géométriques dans la recherche des petits discriminants.) (French) Zbl 0567.12009

Martinet, Jacques

Théorie des nombres, Sémin. Delange-Pisot-Poitou, Paris 1983-84, Prog. Math. 59, 147-179 (1985).

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 \(n\leq 8\).
[For the entire collection see Zbl 0561.00004.]

Reviewer: H. J. Godwin (Egham)

MSC:

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

Keywords:

methods of geometry of numbers; defining polynomial; discriminant; bibliography

Citations:

Zbl 0561.00004