On the class group of an imaginary cyclic field of conductor \(8p\) and \(2\)-power degree
Let \(p=2^{e+1}q+1\) (with \(q\) odd) be a prime and denote by \(K\) the imaginary cyclic subfield of \(\mathbb Q(\zeta_p)\) of degree \(2^{e+1}\). Let \(F\ne K\) be the imaginary subfield of \(K(\sqrt 2)\) containing \(K^+\), the maximal real subfield of \(K\). The authors determine the structure of the \(2\)-class group \(A_F\) of \(F\). Let \(\kappa\) the smallest integer such that the prime \(p\) splits completely in \(\mathbb Q(2^{e-\kappa+1})\). It is shown in Theorem \(1\) that if \(e\ge2\) and \(\kappa\ge1\), then \(A_F\) is the direct sum of \(2^{e-\kappa+1}\) copies of \(\mathbb Z/2\mathbb Z\). The case \(e\ge2\) and \(\kappa=0\) is more complicated and is treated in Theorem 2.
scientific article; zbMATH DE number 7394485
scientific article