Let \(k\) be a real quadratic number field with discriminant \(d_k\) and fundamental unit \(\varepsilon_k\). The \(2\)-class field tower of \(k \;(=k_0)\) is the sequence \(\{k^{(n)}\}_{n\geq 0}\) of fields where \(k^{(n+1)}\) is the maximal abelian unramified \(2\)-extension of \(k^{(n)}\). The narrow \(2\)-class field tower of \(k \;(=k_+^{(0)} \)) is the sequence \(\{k_+^{(n)}\}_{n\geq 0}\) of fields where \(k_+^{(n+1)}\) is the maximal abelian \(2\)-extension of \(k^{(n)}\) unramified outside \(\infty\). The goal of this paper is to compare these two towers for the family of real quadratic fields \(k\) the discriminant \(d_k\) of which is a sum of two squares, and for which the \(2\)-class group is isomorphic to two copies of \(\mathbb Z/2\mathbb Z\). For this last family of real quadratic fields, R. Couture and A. Derhem [C. R. Acad. Sci., Paris, Sér. I 314, No. 11, 785–788 (1992; Zbl 0778.11059)] described the \(2\)-class field tower of \(k\); in particular, they proved that the tower terminates at either \(k^{(1)}\) or \(k^{(2)}\). They used the known fact that if \(H\) is a finite \(2\)-group, then its abelianization is either the Klein group of order \(4\), or the quaternion group of order \(m \geq 8 \) or the dihedral group of order \(2m\geq 8\), or the semi-dihedral group of order \(m\geq 16\). They also proved that the discriminant \(d_k\) is a \mbox{product} \(d_1d_2d_3\) of three positive prime discriminants. Let \(G=\mathrm{Gal}(k^{(2)}/k)\) and let \(G^+=\mathrm{Gal}(k_+^{(2)}/k)\). In the case \(N(\varepsilon_k)=-1\), they wrote the possible structures of \(G\) and \(G^+\) and the corresponding values of the possible Kronecker symbols \((d_i,d_j)\) and the rational biquadratic residue symbols \((d_i,d_j)_4\). In the case \(N(\varepsilon_k)=+1\), they determined \(G\), but not \(G^+\).
The authors first determined the exact values of the integers \(m\) appearing in the results of Couture-Derhem for the values of \(G \) and \(G^+\) in the case where \(N(\varepsilon_k)=-1\). Yet, the major goal of the paper under review, is to obtain some properties of \(G^+\) in the case where \(N(\varepsilon_k)=+1\). The authors exhibited conditions (involving Kronecker symbols) under which the \(2\)-class group of \(k_+^{(1)}\) is not elementary and has rank \(2\) or \(3\). Moreover, they also gave conditions under which the length of the narrow \(2\)-class field tower of \(k\) is either 2 or is \(\geq 3\).
Though it seems that there was “no royal road” to achieve their goals, the authors could count on their expertise to use many arithmetic properties of the number fields involved. For instance, the abelianization of \(\mathrm{Gal}(k^{(2)}/k) \) is the narrow \(2\)-class group of \(k\) and the abelianization of \(\mathrm{Gal}(k_+^{(2)}/k) \) is the narrow \(2\)-class group of \(k\). They appealed to the fundamental units and to the 2-class number of \(\mathbb{Q}(\sqrt{d_1})\) and \(\mathbb{Q}(\sqrt{d_2d_3})\) and also of the compositum of these last two fields. They gave some properties of the commutator subgroup of \(G^+\). They used Kuroda’s class number formula, Hasse’s unit index formula, relative norms of units. They calculated the order of some \(2\)-ambiguous class group and of some capitulation kernel. They calculated decomposition of primes into product of prime ideals in some fields. They derived properties of some Pell equations of the form \(D_1x^2/4^{\delta} -D_2y^2=1\). They used the Ambiguous Class Number formula and the product formula for the Hilbert symbols.