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Galois invariants for local units. (English) Zbl 0858.11060

Let \(K/k\) be a finite Galois extension of \(p\)-adic number fields with Galois group \(G\). In a previous paper [Proc. Lond. Math. Soc., III. Ser. 70, No. 2, 264-284 (1995; Zbl 0828.11062)] the authors proved that the isomorphism class of the \(\mathbb{Z}_pG\)-module \(U_1\) of principle units of \(K\) depends only on the following data: the \(p\)-power roots of unity \(\mu(p)\) contained in \(K\), the degree \([k: \mathbb{Q}_p]\), a certain arithmetically determined \(\mathbb{Z}_p G\)-lattice \(\widehat W\) and the kernel \({\mathcal U}_1\) of a homomorphism \(H^1(G, \operatorname{Hom} (W,\mu (p)) \to H^1 (G, \operatorname{Hom}(W,U))\), where \(U\) is the group of units of \(K\) and \(\widehat W\) is the \(p\)-completion of \(W\).
Here the authors give a way how to decide whether an element \(x\) is contained in \({\mathcal U}_1\). This can be done without the full knowledge of \(U\), using some arithmetic data. They further study the ring structure of \([W,W] = H^0(G, \operatorname{Hom} (W,W))\). In section 6 they describe those summands of \(U_1\) which are not cohomologically trivial.

MSC:

11S23 Integral representations
11S20 Galois theory

Citations:

Zbl 0828.11062
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