The class group of an absolutely abelian \(\ell\)-extension
Let \(\ell\) be an odd prime. In [J. Lond. Math. Soc. 29, 211--217 (1954; Zbl 0055.03302)] and in his book [``Central extensions, Galois groups ad ideal class groups of number fields'', Contemp. Math. 24 (1983; Zbl 0519.12001)], \textit{A. Fröhlich} determined the finite abelian \(\ell\)-extensions \(K\) of \(\mathbb Q\) the class number of which is relatively prime to \(\ell\). In this paper the authors reconsider this problem from a somewhat different point of view. Let \(K/\mathbb Q\) be a finite abelian \(\ell\)-extension and assume \(K\) coincides with the genus field \(K_ G\) of \(K/\mathbb Q\). Let \(K_ C\) denote the central class field of \(K/\mathbb Q\). The result of Fröhlich suggests that the essential point is in the case where \(\mathrm{Gal}(K/\mathbb Q)\) is elementary. The authors first give this reduction as follows: Let \(K_ 1/\mathbb Q\) be the maximal elementary subextension of \(K\). Then: \(\ell\)-rank of \(\mathrm{Gal}(K_ C/K)=\ell\)-rank of \(\mathrm{Gal}(K_{1C}/K_ 1)\) (cf. corollary (1.4) in the paper by \textit{S. V. Ullom} and \textit{S. B. Watt} [Ill. J. Math. 32, No. 3, 422--427 (1988; Zbl 0654.12005)]). Second, assume \(\mathrm{Gal}(K/\mathbb Q)\cong (\mathbb Z/\ell \mathbb Z)^ m\). Using the result \[ \mathrm{Gal}(K_ C/K_ G)\cong \mathrm{coker}(\oplus_{p}\bigwedge^ 2(G_ p)\to \bigwedge^ 2(\mathrm{Gal}(K/\mathbb Q)), \] the authors next investigate the \(\ell\)-rank of \(\mathrm{Gal}(K_ C/K_ G)\), especially in the case \(m\leq 3\), where \(G_ p\) denotes the decomposition group of a prime \(p\) ramified in \(K\). In particular, in the case \(m=3\) they obtain a condition for \(\ell \nmid h_ K\) in terms of decomposition fields of ramified primes in \(K\), where \(h_ K\) denotes the class number of \(K\). Finally they give an algorithm for determining whether \(\ell \mid h_ K\) or not, from which Fröhlich's criterion is deduced.
scientific article; zbMATH DE number 4067043