Real quadratic fields with abelian 2-class field tower (Q1277178)

The authors determine all real quadratic fields \(K\) for which the 2-class field tower has length at most 1, i.e., for which the 2-Hilbert class field has odd class number. By a result of \textit{R. J. Bond} [J. Number Theory 30, 1-10 (1988; Zbl 0649.12008)], the 2-class group of such \(K\) has 2-rank at most 2, and the case of interest is when it is equal to 2. In this case the number \(t\) of distinct prime factors of the discriminant of \(K\) is 3 or 4. For \(t=3\), the classification obtained is in terms of the relative quadratic and biquadratic behavior of these prime factors and the norm of the fundamental unit of \(K\). For \(t=4\), the non-trivial relation between the ideal classes of the ramified primes of \(K\) is also of importance.