Capitulation of 2-class ideals of \(\mathbb Q\left(\sqrt{-pq(2+\sqrt 2)}\right)\) where \(p\equiv q\equiv {\pm 5} \bmod 8\)
This article concerns capitulation of ideals of the field \(K= \mathbb Q(\sqrt{-pq(2+\sqrt{2})})\) when \(p\) and \(q\) are distinct prime numbers with \(p\equiv q\equiv\pm 5\text{\,mod\,}9\). In particular, it discusses how many 2-ideal classes of \(K\) capitulate in the quadratic subextensions of \(K_2^{(1)}/K\), where \(K_2^{(1)}\) is the Hilbert 2-class field of \(K\). This is used to determine the isomorphism type of \(G=\text{Gal}(K^{(2)}_2/K)\), where \(K_2^{(2)}\) is the Hilbert 2-class field of \(K_2^{(1)})\). Along the way, systems of fundamental units for number fields related to \(K\) are found, in order that certain norm groups relevant to the issue of capitulation may be computed. Before summarizing the main results, we first observe that \(\text{Gal}(K_2^{(1)}/K)\) is the non-cyclic group of order 4 -- see \textit{E. Brown} and \textit{C. J. Parry} [ Pac. J. Math. 78, 11--26 (1978; Zbl 0405.12009)], so that \(K_2^{(1)}/K\) contains three proper subextensions \(F_1\), \(F_2\) and \(F_3\), all quadratic over \(K\). Théorème 3.2 states that when \(p\equiv q\equiv 3\bmod 8\), \(K^{(2)}_2= K^{(1)}_2\) so that \(G\) is also non-cyclic of order 4. Further, in this case all four 2-ideal classes of \(K\) capitulate in \(F_1\), \(F_2\) and \(F_3\). Capitulation of 2-ideal classes in \(F_1\), \(F_2\) and \(F_3\) when \(p\equiv q\equiv 5\bmod 8\) is less straightforward and depends upon the unit index of \(K_0= \mathbb Q(\sqrt{2}, \sqrt{pq})\), which is either 1 or 2. The content of Theorem 3.4 is to describe this capitulation in the two cases. Theorem 3.7 then deals with the isomorphism type of \(G\) when \(p\equiv q\equiv 5\bmod 8\), stating that \(G\) is dihedral when the afore-mentioned unit index is 1, and (generalized) quaternion when it is 2. In both cases, \(G\) has order \(2^m\) with \(m\geq 3\). The article concludes by demonstrating that the 2-class group of \(K(\sqrt{pq})\) in the case \(p\equiv q\equiv 5\bmod 8\) is cyclic and has order given in terms of the unit index of \(K_0\) and the 2-class number of \(\mathbb Q(\sqrt{pq})\). Some short examples also follow.
scientific article; zbMATH DE number 5554659