On the 2-component of the class group of certain cyclic extensions of degree \(2^ n\)
For a totally imaginary number field \(K\), cyclic over \(\mathbb{Q}\) of degree \(2^n\) \((n\geq 2)\), the author shows that the 2-class group of \(K\) is cyclic if and only if the relative class number of \(K\) is congruent to \(2\bmod 4\), and in this case, the exponent of this 2-group is \(\leq n\) and is completely determined by the ramification indices of the prime factors of the conductor of \(K\). This contrasts with the case of imaginary quadratic fields, for which the determination of the exponent of the 2-class group -- when it is cyclic -- is still an open problem.
scientific article; zbMATH DE number 792474