On the \(T\)-ramified, \(S\)-split \(p\)-class field towers over an extension of degree prime to \(p\)
Let \(L/K\) be an extension of number fields, let \(N\) be the normal closure of \(L\) over \(K\), let \(p\) be a prime number not dividing \([N:K]\) and let \(T\) and \(S\) be disjoint finite sets of places of \(K\). The \(T\)-ramified, \(S\)-split \(p\)-class field tower \({\bar H_{K,T}^S}(p)\) of \(K\) is the maximal pro-\(p\)-extension of \(K\) unramified outside \(T\) and totally split on \(S\). The aim of the paper under review is to characterize when \[ {\bar H_{K,T}^S}(p)\cdot L=\bar{H}^{S'}_{L,T'}(p), \] \(T'\) and \(S'\) being the sets of places of \(L\) above those of \(T\) and \(S\). In Theorem 1, using a result of Tate, the author shows that a necessary and sufficient condition for the above equality is that \(\mathrm{Gal}(H_{K,T}^S(p)/K)\) and \(\mathrm{Gal}(H_{L,T'}^{S'}(p)/L)\) have the same \(p\)-rank, where \(H_{K,T}^S(p)/K\) (resp. \(H_{L,T'}^{S'}(p)/L\)) is the maximal abelian subextension of \(\bar H_{K,T}^S(p)/K\) (resp. \(\bar H_{L,T'}^{S'}(p)/L\)). Using this result and some class field theory, in Theorem 2 a characterization in terms of ideal classes and units of \(L\) is obtained. The last part of the article contains some examples and applications.
scientific article; zbMATH DE number 5604008
scientific article