The $4$-rank of class groups of $K(\sqrt{n})$

Abstract: Let $K/\mathbb{Q}$ be a quadratic extension. In this paper we study the $4$-rank of the class group $\text{Cl}(K(\sqrt{n}))$, where $n$ varies over squarefree rational integers. We show that for $100\%$ of squarefree $n$, the $4$-rank is given by an explicit formula involving the $2$-rank of $\text{Cl}(K)$ and the number of prime factors of $n$ which are inert in $K/\mathbb{Q}$.