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Linear theories of global fields with absolute values

Arno Fehm, Pierre Touchard

Abstract

We study the theory of a global field k as a k-vector space with a predicate for one of the absolute values on k. For example, we prove that in this language a global field with an ultrametric or real archimedean absolute value has a decidable theory, while with a complex absolute value the theory is always undecidable. We also study the existential theories and axiomatize k together with predicates for all non-complex absolute values on k simultaneously.

Linear theories of global fields with absolute values

Abstract

We study the theory of a global field k as a k-vector space with a predicate for one of the absolute values on k. For example, we prove that in this language a global field with an ultrametric or real archimedean absolute value has a decidable theory, while with a complex absolute value the theory is always undecidable. We also study the existential theories and axiomatize k together with predicates for all non-complex absolute values on k simultaneously.

Paper Structure

This paper contains 8 sections, 29 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Places and vector spaces
  3. Quantifier elimination for discretely valued vector spaces
  4. Absolute values
  5. Ultrametric absolute values
  6. Real absolute values
  7. Complex absolute values
  8. Weak approximation

Key Result

Theorem 1.1

Let $\mathfrak{p}\in\mathbb{P}_k$ be a place of a global field $k$.

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 61 more