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Structure galoisienne et forme trace. (Galois structure and trace form). (French) Zbl 0659.12013
Sémin. Théor. Nombres, Univ. Bordeaux I 1986-1987, Exp. No. 24, 6 p. (1987).
There exists an extensive literature on the link between Galois structure and ramification in number fields. A new type of results in this spirit is given here, without proofs. The inverse different of a normal extension K/\({\mathbb{Q}}\) with Galois group G, of odd degree, is a square of an integral ambiguous ideal \(A_ K\). A number of results are given on the relationship between the following conditions on K:
(i) All second ramification groups of K/\({\mathbb{Q}}\) are trivial;
(ii) \(A_ K\) is a ‘sum of squares’ with respect to its structure of \({\mathbb{Z}}\)-lattice equipped with the quadratic form T given by the absolute trace;
(iii) \(A_ K\) is a free (resp. a projective) module over the integral group-ring \({\mathbb{Z}}G.\)
Finally, an explicit calculation in a special case is mentioned which describes the structure of \((A_ K,T)\) as essentially the Niemeier lattice associated to the root system \(A^ 6_ 4\).
Reviewer: J.Brinkhuis

11R32 Galois theory
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
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