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Sur les réseaux unimodulaires pour $$Trace(X^ 2)$$. (On unimodular lattices for $$Trace(X^ 2))$$. (French) Zbl 0734.11029
Sémin. Théor. Nombres, Paris/Fr. 1988-89, Prog. Math. 91, 1-12 (1990).
[For the entire collection see Zbl 0711.00009.]
If K is a number field then each ideal of K is a lattice for the quadratic form $$T_{K/{\mathbb{Q}}}(x^ 2)$$ where $$T_{K/{\mathbb{Q}}}$$ is the usual trace function. An ideal I is unimodular for $$T_{K/{\mathbb{Q}}}$$ if and only if $$I^ 2={\mathcal D}_ K^{-1}$$ where $${\mathcal D}_ K$$ is the different for K/$${\mathbb{Q}}$$. In particular, if [K : $${\mathbb{Q}}]$$ is odd there exists a unique ideal $$A_ K$$ such that $$A^ 2_ K={\mathcal D}_ K^{- 1}.$$
Assume from now on that K/$${\mathbb{Q}}$$ is a normal abelian extension of odd degree with Galois group G. If K/$${\mathbb{Q}}$$ is cyclic of degree $$p^ n$$ with p totally ramified in K, Theorem 1 gives a decomposition of $$A_ K$$ as an orthogonal sum of $${\mathbb{Z}}$$-indecomposable $${\mathbb{Z}}[G]$$ unimodular lattices. Theorem 2 assumes only the weaker hypothesis that the ramification index of each prime p is a power of p. It is then shown that $$A_ K$$ is $${\mathbb{Z}}[G]$$ isometric to a direct sum of tensor products of lattices corresponding to the inertia subgroups $$I_ p$$ of the primes p with ramification indices divisible by $$p^ 2$$.

##### MSC:
 11E04 Quadratic forms over general fields 11R20 Other abelian and metabelian extensions