On the rank of the 2-class group of an extension of degree 8 over $\mathbb{Q}$ (Q2416507)

Consider an imaginary cyclic quartic number field $K$ with nontrivial $2$-class group. The objective of this paper is to compute the rank of the $2$-class group of $F$ where $F/K$ is an unramified quadratic extension. The authors describe an arithmetic method to compute this rank. The field $K$ under consideration is given as follows. Consider $p$ and $l$ different primes such that $l\equiv 5 \bmod 8$, $p\equiv 1\bmod 4$, $\big(\frac pl\big)=1$ and $\big(\frac 2p\big)=-\big(\frac pl\big)_4=\big(\frac lp \big)_4$ where $\big(\frac dp \big)_4$ denotes the rational biquadratic symbol. Let $\varepsilon$ be the fundamental unit of $k={\mathbb{Q}} (\sqrt{l})$. Let $K=\big(\sqrt{-2p \varepsilon \sqrt{l}}\big)$ and $F=K(\sqrt{p})$. Let $C_2$ denote the cyclic group of order $2$. \par The main theorem describes explicitly the $2$-class group of $K$ which is isomorphic to $C_2\times C_2\times C_2$, states that the $2$-class number of $F$ is $8$ and gives the $2$-class group of $F$ which is isomorphic to $C_2\times C_2\times C_2$. In particular, the rank of the $2$-class group of $F$ is $3$.