\(\Gamma\)-extensions and corestriction homomorphism
Let \(k\) be a number field, \(p\) a prime and \(S\) a finite set of places of \(k\), containing all divisors of \(p\) in \(k\). If \(p=2\) we assume that \(k\) is totally imaginary. Let \(k_ S\) be the maximal \(p\)-extension of \(k\) unramified outside \(S\) and let \(K/k\) be a \(\Gamma\)-extension. Then \(k_ n\) denotes the unique subextension of \(K/k\) of degree \(n\). The field \(K\) is contained in \(k_ S\). The main result of the author is the following Stabilisation Theorem: The relation rank \(r(G(k_ S/k_{n+1}))\) is greater or equal \(r(G(k_ S/k_ n))\). The group \(G(k_ S/K)\) is a free pro-\(p\)-group if and only if \(r(G(k_ S/k_{N+1}))= r(G(k_ S/k_ N))\) for some \(N\). If such an \(N\) exists, then \(r(G(k_ S/k_ n))= r(G(k_ S/k_ N))\) for all \(n\geq N\).
scientific article; zbMATH DE number 404569