On trace-one generators of abelian cubic fields

TL;DR

This paper establishes a precise relationship between trace-one monic integral polynomials generating a specific abelian cubic field and ideals in a quadratic field, linking polynomial roots to ideal norms.

Contribution

It provides a new explicit formula connecting the count of trace-one polynomials to ideals in a quadratic field for tamely ramified abelian cubic fields.

Findings

Number of trace-one polynomials equals the count of ideals with specific norm

Establishes a direct link between polynomial roots and quadratic field ideals

Results apply to tamely ramified abelian cubic fields

Abstract

Let $K$ be a tamely ramified abelian cubic number field with discriminant $D_K$. We prove that the number of trace-one monic integral polynomials with root field $K$ and height $H$ is equal to the number of ideals in the quadratic field $\mathbb Q(\sqrt{-3})$ with norm $H^2 D_K^{-1/2}$.