Wingberg, Kay On the maximal unramified \(p\)-extension of an algebraic number field. (English) Zbl 0779.11054 J. Reine Angew. Math. 440, 129-156 (1993). 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 \(p<125000\). Reviewer: J.Brinkhuis (Rotterdam) Cited in 11 Documents MSC: 11R32 Galois theory Keywords:Galois group; maximal unramified \(p\)-extension; finitely generated free pro-\(p\)-group; Vandiver’s conjecture PDFBibTeX XMLCite \textit{K. Wingberg}, J. Reine Angew. Math. 440, 129--156 (1993; Zbl 0779.11054) Full Text: DOI Crelle EuDML