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On the maximal unramified \(p\)-extension of an algebraic number field. (English) Zbl 0779.11054

It is shown that under certain conditions the Galois group \(G(L/k_ \infty)\) of the maximal unramified \(p\)-extension \(L\) of the cyclotomic \(\mathbb{Z}_ p\)-extension \(k_ \infty\) of \(k\) is a finitely generated free pro-\(p\)-group. If \(k=\mathbb{Q}(\varphi_ p)\), where \(\varphi_ p\) is a primitive \(p\)th root of unity, this condition for freeness of \(G(L/k_ \infty)\) is just Vandiver’s conjecture, and so \(G(L/\mathbb{Q}(\varphi_{p\infty}))\) is free at least for \(p<125000\).

MSC:

11R32 Galois theory
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