Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on Diophantine subsets of large fields

Andrew Kwon

Abstract

Large fields (also called ample, anti-mordellic) generalize many fields of classical interest, such as algebraically closed fields, real-closed fields, and $p$-adic fields. In this note we answer a question of Pop by generalizing a result of Fehm and prove that finite unions of affine translates of infinite proper subfields are never diophantine subsets of perfect large fields.

A note on Diophantine subsets of large fields

Abstract

Large fields (also called ample, anti-mordellic) generalize many fields of classical interest, such as algebraically closed fields, real-closed fields, and -adic fields. In this note we answer a question of Pop by generalizing a result of Fehm and prove that finite unions of affine translates of infinite proper subfields are never diophantine subsets of perfect large fields.

Paper Structure

This paper contains 5 sections, 11 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Model-theoretic background
  3. Images of smooth morphisms
  4. The main reduction
  5. Final comments and questions

Key Result

Theorem 1

(Fe10, Theorem 2) Let $k$ be a perfect large field and $\Sigma \subset k$ be an infinite diophantine subset. Then for every proper subfield $k_{0} \subset k$, $|\Sigma \setminus k_{0}| = |k|$.

Theorems & Definitions (13)

  • Theorem
  • Theorem 1.1
  • Lemma 1.2
  • Theorem 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 3.1
  • proof
  • Lemma 4.1: Fe10, Lemma 8
  • ...and 3 more