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On the existential theory of the completions of a global field

Philip Dittmann, Arno Fehm

Abstract

We discuss the common existential theory of all or almost all completions of a global function field.

On the existential theory of the completions of a global field

Abstract

We discuss the common existential theory of all or almost all completions of a global function field.
Paper Structure (6 sections, 26 theorems, 12 equations)

This paper contains 6 sections, 26 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Existential $\mathfrak{L}_{\rm val}$-theory without parameters
  4. Existential $\mathfrak{L}_{\rm val}$-theory with constant parameters, and the perfect hull
  5. Existential $\mathfrak{L}_{\rm ring}$-theory with parameters
  6. Existential $\mathfrak{L}_{\rm val}$-theory with parameters

Key Result

Theorem 1.1

The (common) $\mathfrak{L}_{\rm val}$-theory of all $\mathbb{Q}_p$ and the $\mathfrak{L}_{\rm val}$-theory of almost all $\mathbb{Q}_p$ are both decidable.

Theorems & Definitions (55)

  • Theorem 1.1: Ax
  • Corollary 1.2
  • Theorem 1.3: AF23
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Theorem 3.1: AF16
  • Proposition 3.2
  • proof
  • ...and 45 more