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On the Northcott property for infinite extensions

Martin Widmer

Abstract

We start with a brief survey on the Northcott property for subfields of the algebraic numbers $\Qbar$. Then we introduce a new criterion for its validity (refining the author's previous criterion), addressing a problem of Bombieri. We show that Bombieri and Zannier's theorem, stating that the maximal abelian extension of a number field $K$ contained in $K^{(d)}$ has the Northcott property, follows very easily from this refined criterion. Here $K^{(d)}$ denotes the composite field of all extensions of $K$ of degree at most $d$.

On the Northcott property for infinite extensions

Abstract

We start with a brief survey on the Northcott property for subfields of the algebraic numbers . Then we introduce a new criterion for its validity (refining the author's previous criterion), addressing a problem of Bombieri. We show that Bombieri and Zannier's theorem, stating that the maximal abelian extension of a number field contained in has the Northcott property, follows very easily from this refined criterion. Here denotes the composite field of all extensions of of degree at most .
Paper Structure (3 sections, 7 theorems, 20 equations)

This paper contains 3 sections, 7 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Silverman's inequality
  3. Theorem \ref{['prop:1']} implies Bombieri and Zannier's Theorem \ref{['thm: 1']}

Key Result

Theorem 1

Given a number field $K$, $n\in \mathbb N$, and $X\geq 1$, there are only a finite number of points $P$ in $\mathbb P^n(K)$ such that $H(P)\leq X$.

Theorems & Definitions (14)

  • Theorem 1: Northcott, 1950
  • Definition 1: Northcott property
  • Theorem 2: Northcott's Theorem
  • Theorem 3: Bombieri, Zannier 2001
  • Theorem 4: WidmerPropN
  • Theorem 5
  • proof
  • Lemma 1: Silverman, 1984
  • proof
  • Lemma 2: Bombieri and Zannier BoZa
  • ...and 4 more