On Dirichlet biquadratic fields

TL;DR

This paper investigates the 4-rank of the ideal class group in certain biquadratic fields, establishing a relation to prime divisors of the integer parameter for a positive proportion of cases.

Contribution

It provides a new explicit formula for the 4-rank of class groups in Dirichlet biquadratic fields for a significant set of integers.

Findings

The 4-rank equals the number of prime divisors congruent to 3 mod 4 minus one.

A positive proportion of squarefree integers n satisfy this 4-rank relation.

The result links prime factorization properties to class group structure.

Abstract

We study the $4$-rank of the ideal class group of $K_n := \mathbb{Q}(\sqrt{-n}, \sqrt{n})$. Our main result is that for a positive proportion of the squarefree integers $n$ we have that the $4$-rank of $\text{Cl}(K_n)$ equals $\omega_3(n) - 1$, where $\omega_3(n)$ is the number of prime divisors of $n$ that are $3$ modulo $4$.