On the Kummer radical of \(\mathbb Z_\ell\)-extensions (Q2833598)

If \(p\) is an odd prime and the algebraic number field \(K\) contains the \(p\)-th roots of unity, one has Kummer theory for abelian extensions \(L/K\) of degree \(p\), and it is an old and difficult question which of these \(L\) start a \(\mathbb Z_p\)-extension, that is, are contained in the compositum \(M\) of all \(\mathbb Z_p\)-extensions of \(K\). The Galois group of \(M/K\) is \(\mathbb Z_p^d\) where \(d\geq r_2(K)+1\) (Leopoldt's conjecture stating that this is an equality), but often there are more than \(d\) independent abelian \(p\)-ramified extensions \(L/K\) of degree \(p\). The paper under review gives a very elegant description of extensions \(L/K\) that start a \(\mathbb Z_p\)-extension in terms of Kummer theory and projective limits (Theorem 6). The question remains how accessible this description is in concrete cases.NEWLINENEWLINEThe main theorem of the present article strengthens previous results in three papers of \textit{S. Seo} (the first one being ``On first layers of \(\mathbb Z_p\)-extensions'' [J. Number Theory 133, No. 12, 4010--4023 (2013; Zbl 1318.11143)]; for the others see the references in the author's paper), and it also contains analogous results concerning the Tate kernel attached to \(K_2\) of number fields.NEWLINENEWLINEThe paper is very clearly written. One small comment: the author points out, rightly, that Seo's papers lack references to some important earlier work. In a similar vein, in the paper under review the early paper by \textit{J. E. Carroll} and \textit{H. Kisilevsky} [``Initial layers of \(\mathbb Z_\ell\)-extensions of complex quadratic fields'', Compos. Math. 32, 157--168 (1976; Zbl 0357.12003)] might perhaps have been mentioned. Even if the methods of that paper are quite different and its results are much more special than those in the paper under review, it was a kind of start.