On regular number fields
Fix a prime number \(\ell\) and let \(\zeta\) denote a primitive \(\ell\)-th root of unity. Let \(K\) be a number field with divisor group \(D_ K\) and write \(\mathcal D_ K\) for the multiplicative tensor product \(\mathbb Z_{\ell}\otimes_{\mathbb Z}D_ K\). One can define a surjective map (called Gras's logarithm), \(\text{lg}: \mathcal D_ K\to \text{Gal}(Z/K)\), where \(Z\) is the composite field of the \(\mathbb Z_{\ell}\)-extensions of \(K\). For a finite or a real place \(\wp\) of \(K\), \(\text{lg}(\wp)=1\) whenever \(\wp\) is real or lies over \(\ell\), otherwise \(\text{lg}(\wp)\) is a topological generator of the decomposition group \(D_{\wp}(Z/K)\simeq\mathbb Z_{\ell}\) associated to \(\wp\) in the abelian extension \(Z/K\). A finite set \(S\) of places of \(K\) is called primitive when the \(\text{lg}(s)\), \(s\in S\), form a \(\mathbb Z_{\ell}\)-basis of a pure submodule of \(\text{Gal}(Z/K)\). An \(\ell\)-extension \(L/K\) is called primitively ramified if the set \(S\) of places of \(K\) that ramify tamely in \(L/K\), is primitive. \(K\) is called regular (with respect to \(\ell\)) if the \(\ell\)-Sylow subgroup \(R_ 2(K)\) of the kernel in \(K_ 2(K)\) of the regular symbols attached to the non-complex places of \(K\), is trivial. When \(K\) contains the maximal real subfield \(k=\mathbb Q(\zeta +\zeta^{-1})\) of the cyclotomic field \(\mathbb Q(\zeta)\) several equivalent characterizations of regularity can be given, one of which says that \(K\) is regular if and only if \(K\) verifies Leopoldt's conjecture (with respect to \(\ell)\) and the torsion submodule \({\mathcal T}_ K\) of \(\text{Gal}(M/K)\), with \(M\) the maximal \(\ell\)-ramified, abelian \(\ell\)-extension of \(K\) which decomposes completely at the infinite places, is zero. A classical example of a regular number field is provided by the cyclotomic field \(\mathbb Q(\zeta)\) if and only if \(\ell\) is a regular prime in the usual terminology. Writing \(\delta_ K\) for the defect of Leopoldt's conjecture for \(K\), \(K\) is called an \(\ell\)-rational field if \({\mathcal T}_ K=\{0\}\) and \(\delta_ K=0\). The main results of the paper can now be formulated: 1) Let \(K\) contain the maximal real subfield \(k\), and let \(L/K\) be a Galois \(\ell\)-extension. Then the following conditions are equivalent: (i) \(L\) is regular; (ii) \(K\) is regular and \(L/K\) is primitively ramified. 2) Let \(L/K\) be a Galois \(\ell\)-extension. Then the following conditions are equivalent: (i) \(L\) is \(\ell\)-rational; (ii) \(K\) is \(\ell\)-rational and \(L/K\) is primitively ramified.
scientific article; zbMATH DE number 4154554