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Galois module structure of algebraic integers of cyclic cubic fields

Miho Aoki

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 $L_n=\mathbb Q(ρ_n)$ be a cyclic cubic field with Galois group $G:={\rm Gal}(L_n/\mathbb Q)$, where $ρ_n$ 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 $f_n(X)$.

Galois module structure of algebraic integers of cyclic 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 . Let be a cyclic cubic field with Galois group , where is a root of , and the ring of integers of . We explicitly give the generator of the free module of rank over the associated order by using the roots of .

Paper Structure

This paper contains 6 sections, 17 theorems, 143 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Integral Basis
  3. Branch classes and associated orders
  4. Tamely ramified cases
  5. Wildly ramified cases
  6. Examples

Key Result

Theorem 1

Let $n=n_1/n_2$ be a rational number where the integers $n_1$ and $n_2$ are coprime. Suppose that the cubic polynomial $f_n(X)$ is irreducible over $\mathbb Q$, and $3\nmid n_1$ or $n_1=3t\, (t\in \mathbb Z)$, $n_2 \equiv -2t \pmod{9}$. There exist integers $a_0$ and $a_1$ that satisfy $ec=a_0^2-a_0 where $\left( \frac{\cdot}{3} \right)$ is the Legendre symbol, and is a generator of a normal inte

Theorems & Definitions (29)

  • Theorem 1: =Theorem \ref{['theo:tame']}
  • Theorem 2: =Theorem \ref{['theo:wild']}
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 19 more