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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

##### MSC:
 11R32 Galois theory 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
##### Keywords:
Galois structure; ramification; sum of squares; trace
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