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Small generators of abelian number fields

Martin Widmer

Abstract

We show that for each abelian number field $K$ of sufficiently large degree $d$ there exists an element $α\in K$ with $K=\IQ(α)$ and absolute Weil height $H(α)\ll_d |Δ_K|^{1/2d}$ , where $Δ_K$ denotes the discriminant of $K$. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent $1/2d$ is best-possible when $d$ is even.

Small generators of abelian number fields

Abstract

We show that for each abelian number field of sufficiently large degree there exists an element with and absolute Weil height , where denotes the discriminant of . This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent is best-possible when is even.
Paper Structure (4 sections, 7 theorems, 47 equations)

This paper contains 4 sections, 7 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Primes of a given residue class in short intervals
  3. Proof of Theorem \ref{['Thm:2']}
  4. Proof of Theorem \ref{['Thm:1']}

Key Result

Theorem 1

Let $L$ be the effectively computable constant from (def:nuexplicit), and suppose $d\geq 4L$. Then, apart from finitely many exceptional fields, we have for each abelian number field $K$ of degree $d$.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • Proposition 2
  • proof