## 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

### MSC:

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

Zbl 0561.00004