Journal of the
Korean Mathematical Society JKMS

For imaginary quadratic number fields $F=\mathbb Q(\sqrt{\varepsilon p_1\cdots p_{t-1}})$, where $\varepsilon \in \{- 1,- 2\}$ and distinct primes $p_i\equiv 1 \bmod 4$, we give conditions of $8$-ranks of class groups $C(F)$ of $F$ equal to $1$ or $2$ provided that $4$-ranks of $C(F)$ are at most equal to $2$. Especially for $F=\mathbb Q(\sqrt{\varepsilon p_1 p_2})$, we compute densities of $8$-ranks of $C(F)$ equal to $1$ or $2$ in all such imaginary quadratic fields $F$. The results are stated in terms of congruence relations of $p_i$ modulo $2^n$, the quartic residue symbol $(\frac{p_1}{p_2})_4$ and binary quadratic forms such as $p_{2}^{h_{+}(2p_1)/4}=x^2-2p_1y^2$, where $h_{+}(2p_1)$ is the narrow class number of $\mathbb Q(\sqrt{2p_1})$. The results are also very useful for numerical computations.

Keywords: class group, unramified extension, quartic residue, density

MSC numbers: 11R29, 11R45