This is a very nice and carefully written paper about the so-called capitulation problem for the ideals of cyclic quartic field \(K\) of the form \(K=\mathbb{Q}(\sqrt{\ell},\sqrt{-p\varepsilon \sqrt{\ell}})\) and of the form \(K=\mathbb{Q}(\sqrt{\ell},\sqrt{-\varepsilon \sqrt{\ell}})\), where \(p\) and \(\ell\) are primes under certain restrictions and where \(\varepsilon\) is the fundamental unit of the quadratic field \(\mathbb{Q}(\sqrt{\ell})\). It is well known that an ideal \(J\) of \(K\) becomes a principal ideal in the Hilbert class field \(H_1\) of \(K\). The capitulation problem consists in investigating if the ideal \(J\) becomes a principal ideal in a proper subfield of \(H_1\). The authors concentrate on \(2\)-ideal classes and assume that the \(2\)-class group of \(K\) is of type \((2,4)\). If \(H_2\) is the class field of \(H_1\), the authors also determine the structure of the Galois group of the extension \(H_ 2\) of \(K\).
Many preliminary results of different authors are used to achieve the proofs.