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A normal basis theorem for infinite Galois extensions. (English) Zbl 0569.12013

Let \(L/K\) be a Galois extension of fields with (profinite) Galois group \(G\), and let \((G,K)\) denote the \(K\)-vector space of all continuous functions \(G\to K\), where \(G\) (resp. \(K\)) is endowed with the Krull (resp. the discrete) topology. The group \(G\) acts on \((G,K)\) by left translations.
Theorem 1. There exists an isomorphism \((G,K)\to L\) of \(K\)-vector spaces that respects the action of \(G\).
Next, let U denote the set of open normal subgroups of \(G\). If \(N,N'\in U\) and \(N\subset N'\), the group homomorphism \(G/N\to G/N'\) induces a homomorphism \(K[G/N]\to K[G/N']\) of the group rings. Let \(K[[G]]=\lim_{N{\overset \leftarrow \in}U}K[G/N]\), the limit being taken with respect to these homomorphisms. For each \(N\in U\) the fixed field \(L^ N\) of N is a finite Galois extension of \(K\) with group \(G/N\). If \(N\subset N'\) we have a trace map \(Tr_{N'/N}: L^ N\to L^{N'}\), defined by \(Tr_{N'/N}(y)=\sum_{\sigma \in N'/N}\sigma(y)\). The inverse limit \(\lim_{N{\overset \leftarrow \in}U}L^ N\) relative to these homomorphisms is naturally a \(K[[G]]\)-module.
Theorem 2. \(\lim_{N{\overset \leftarrow \in}U}L^ N\) is a free \(K[[G]]\)-module of rank one.
When \(L/K\) is finite Galois, both theorems reduce to the normal basis theorem.
Reviewer: I.G.Macdonald

MSC:

12F10 Separable extensions, Galois theory
20E18 Limits, profinite groups
20F29 Representations of groups as automorphism groups of algebraic systems
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