The Capitulation Problem for Certain Cyclic Quartic Number Fields

Abstract

Let $K$ be a cyclic quartic number field such that its 2-class group is of type $(2,4)$, $K_2^{(1)}$ be the Hilbert 2-class field of $K$, $K_2^{(2)}$ be the Hilbert 2-class field of $K_2^{(1)}$ and $G=\text{Gal}(K_2^{(2)}/K)$ be the Galois group of $K_2^{(2)}/K$. Our goal is to study the capitulation problem of 2-ideal classes of $K$ and to determine the structure of $G$.