Normal extension of quartic fields with the symmetric group (Q2646035)

Let \(B/R\) be a normal sextic field with Galois group \(G(B/R) = S_3\), where \(R\) is the rational field and \(S_n\) is isomorphic to the symmetric group on \(n\) letters. Let \(u_1, u_2\) be elements of \(B\) such that \(u_1, u_2\) and \(u_1u_2\) are not squares of elements of \(B\). Then \(N = B(\sqrt{u_1}, \sqrt{u_2})\) is normal over \(B\), \(G(N/ B) = V\) (Vierergruppe), and hence is the class field over \(B\) for an ideal group \(H\) in \(B\) such that \(\mathfrak A =\mathfrak H+\mathfrak C_1+\mathfrak C_2+\mathfrak C_1\mathfrak C_2\), where \(\mathfrak A\) is the group of ideals of \(B\) prime to the conductor of \(\mathfrak H\). \(N\) is normal over \(R\) if and only if \(\mathfrak H^\rho = \mathfrak H\) for every \(\rho\) in \(S_3\) and then there are four possibilities for \(G(N/R)\). The author shows that \(G(N/R) = S_4\) if and only if, for suitable generators \(\sigma\) and \(\tau\), \(\sigma^2=\tau^2=1\) of \(S_3\), \(\mathfrak C_1^\sigma = \mathfrak C_1\mathfrak C_2\), \(\mathfrak C_1^\tau= \mathfrak C_2\), \(\mathfrak C_2^\sigma = \mathfrak C_2\), \(\mathfrak C_2^\tau = \mathfrak C_1\mathfrak C_2\), that is (Kummer theory) if and only if \(u_1^\sigma = u_1u_2a_1^2\), \(u_1^\tau = u_2a_2^2\), \(u_2^\sigma = u_2a_3^2\), \(u_2^\tau = u_1 u_2 a_4^2\), \(a_i\) in \(B\). In case the quadratic subfield of \(B\) is imaginary, there exist units \(u_1, u_2\) of \(B\) satisfying these conditions. Existence theorems of \textit{H. Hasse} for the \(B\) [Math. Z. 31, 565--582 (1930; JFM 56.0167.02)] are combined with the above to obtain \(N\) which give realizations of many of the Hilbert subgroup series of the author's tables [Ann. Math. (2) 38, 739--749 (1937; Zbl 0017.19502)]. Realizations of other entries are given by numerical examples, while the existence of the remaining entries is undecided. The tables are completed by the addition of five series.