Normal integral basis and ray class group modulo 4. (Q595960)

Let \(K\) be a number field. In this note, the question is: under what condition(s) does (do) the following property hold for \(K\): For any tame abelian extensions \(N_1\) and \(N_2\) over \(K\) of exponent 2, the compositum \(N_1N_2/K\) has a relative normal integral basis (NIB) if both \(N_1/K\) and \(N_2/K\) have a NIB? The answer of the authors is: if and only if the ray class group of \(K\) defined modulo 4 is trivial. As a consequence, this solves another version of the question by the second author [J. Number Theory 101, No. 1, 131--137 (2003; Zbl 1027.11083)]. The proof uses classical ingredients with genus theory.