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Gaussian periods and Shanks' cubic polynomials. II

Miho Aoki

TL;DR

The paperAddresses the problem of relating Gaussian periods of a cyclic cubic field to roots of Shanks' cubic polynomials in the wildly ramified setting, extending prior tamely ramified results. It develops a conductor-based parametrization via $n=n_1/n_2$ with $4\mathfrak f=M^2+27N^2$, proves irreducibility and conductor properties for the associated cubic fields $L_n$, and establishes a precise linear relation $n_2^3 \mu(\mathfrak f/9) f_n(X)=P(\mu(\mathfrak f/9)(n_2X-n_1/3))$ that ties Gaussian periods to roots of $f_n$. A key outcome is that all cyclic cubic fields with conductor $\mathfrak f$ arise from suitable $(M,N)$ and that the Gaussian periods $\{\eta_0,\eta_1,\eta_2\}$ correspond (up to the factor $\mu(\mathfrak f/9)$) to the roots of the Shanks polynomial under an explicit affine transformation. The work also analyzes the unit-group structure of the associated order and provides explicit examples, offering a unified framework for understanding period relations across tame and wild ramification with potential computational implications.

Abstract

We give a linear relation between a cubic Gaussian period and a root of Shanks' cubic polynomial in wildly ramified cases.

Gaussian periods and Shanks' cubic polynomials. II

TL;DR

The paperAddresses the problem of relating Gaussian periods of a cyclic cubic field to roots of Shanks' cubic polynomials in the wildly ramified setting, extending prior tamely ramified results. It develops a conductor-based parametrization via with , proves irreducibility and conductor properties for the associated cubic fields , and establishes a precise linear relation that ties Gaussian periods to roots of . A key outcome is that all cyclic cubic fields with conductor arise from suitable and that the Gaussian periods correspond (up to the factor ) to the roots of the Shanks polynomial under an explicit affine transformation. The work also analyzes the unit-group structure of the associated order and provides explicit examples, offering a unified framework for understanding period relations across tame and wild ramification with potential computational implications.

Abstract

We give a linear relation between a cubic Gaussian period and a root of Shanks' cubic polynomial in wildly ramified cases.

Paper Structure

This paper contains 5 sections, 8 theorems, 50 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structure of the units group of the associated order
  4. Proof of the theorem
  5. Examples

Key Result

Theorem 1

Let $\mathfrak f =p_1\cdots p_{\nu}$ be an integer where $p_1,\ldots,p_{\nu}$ are different prime numbers which satisfy $p_1\equiv \cdots \equiv p_{\nu} \equiv 1\pmod{3}$, and a pair $(M,N) \in \mathbb Z \times \mathbb Z$ satisfies (eq:4f). Put $n_1=(M-3N)/2,\ n_2=N$ and $n=n_1/n_2$. Then $n_1$ and Furthermore, all cyclic cubic fields with conductor $\mathfrak f$ are given by $L_n$ for such $n=n_

Theorems & Definitions (14)

  • Theorem 1: A2
  • Theorem 2
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4: AF
  • proof
  • ...and 4 more