Corps de nombres de degré 4 de type alterné. (Number fields of degree four of alternating type). (French) Zbl 0564.12007

The paper summarizes, without proofs, the results obtained by the author in ”Extensions à groupe de Galois \(A_ 4''\) [Thèse de 3e cycle, Bordeaux, 1984]. Let K be a number field of degree 4 over \({\mathbb{Q}}\) whose Galois closure N has Galois group isomorphic to the alternating group \(A_ 4\). The author determines the discriminant of K in terms of the discriminant of the cubic subfield k of N and gives a way to obtain a primitive element of K and its irreducible polynomial. The number of fields K with a given discriminant is determined, if the class number of k is odd. Finally, a table of the fields K with discriminant \(\leq 250^ 2\) and tables of totally real fields K with small discriminant are given.
Reviewer: N.Vila


11R16 Cubic and quartic extensions
11R32 Galois theory
20F29 Representations of groups as automorphism groups of algebraic systems