Galois module structure of algebraic integers of cyclic cubic fields

TL;DR

This paper explicitly determines the Galois module structure of the ring of integers in all cyclic cubic fields generated by roots of a specific polynomial, providing explicit generators over the associated order.

Contribution

It introduces a method to explicitly compute generators of the ring of integers as modules over the associated order in cyclic cubic fields.

Findings

Explicit generators for the ring of integers over the associated order

Application to all cyclic cubic fields generated by roots of the polynomial

Enhanced understanding of Galois module structure in cubic fields

Abstract

We determine the Galois module structure of the ring of integers for all cubic fields using roots of the generic cyclic cubic polynomial $f_n(X)=X^3-nX^2-(n+3)X-1$. Let $L_n=\mathbb Q(\rho_n)$ be a cyclic cubic field with Galois group $G:={\rm Gal}(L_n/\mathbb Q)$, where $\rho_n$ is a root of $f_n (X)$, and ${\mathcal O}_{L_n}$ the ring of integers of $L_n$. We explicitly give the generator of the free module ${\mathcal O}_{L_n}$ of rank $1$ over the associated order ${\mathcal A}_{L_n/\mathbb Q}:= \{ x\in \mathbb Q [G] \, |\, x\, {\mathcal O}_{L_n} \subset {\mathcal O}_{L_n} \}$ by using the roots of $f_n(X)$.