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

TL;DR

This paper provides an explicit formula for the 4-rank of class groups of quadratic extensions of quadratic fields, depending on the 2-rank and inert primes, valid for almost all squarefree integers n.

Contribution

It establishes a precise, explicit relationship for the 4-rank of class groups in quadratic extensions, extending understanding of class group structure in these fields.

Findings

For 100% of squarefree n, the 4-rank is determined by an explicit formula.

The 4-rank depends on the 2-rank of Cl(K) and inert prime factors of n.

The result applies to almost all squarefree integers 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 $K/\mathbb{Q}$.