Class Number Formulas for Certain Biquadratic Fields

TL;DR

This paper derives explicit class number formulas for certain biquadratic fields, using elementary methods and expansions related to primes and units, providing new tools for understanding class numbers in algebraic number theory.

Contribution

It introduces two novel elementary formulas for class numbers of specific biquadratic fields, expanding the computational and theoretical understanding of these invariants.

Findings

Derived class number formulas as sums of rational function coefficients

Expressed class numbers using base-$ ext{ε}_F$ expansions of $1/p$

Provided explicit formulas for fields with class number one

Abstract

We consider the class numbers of imaginary quadratic extensions $F(\sqrt{-p})$, for certain primes $p$, of totally real quadratic fields $F$ which have class number one. Using seminal work of Shintani, we obtain two elementary class number formulas for many such fields. The first expresses the class number as an alternating sum of terms that we generate from the coefficients of the power series expansions of two simple rational functions that depend on the arithmetic of $F$ and $p$. The second makes use of expansions of $1/p$, where $p$ is a prime such that $p \equiv 3 \pmod{4}$ and $p$ remains inert in $F$. More precisely, for a generator $\varepsilon_F$ of the totally positive unit group of $\mathcal{O}_F$, the base-$\varepsilon_{F}$ expansion of $1/p$ has period length $\ell_{F,p}$, and our second class number formula expresses the class number as a finite sum over disjoint cosets of size $\ell_{F,p}$.