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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