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Universal quadratic forms and indecomposables in number fields: A survey

Vítězslav Kala

Abstract

We give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discuss indecomposable algebraic integers as one of the main tools.

Universal quadratic forms and indecomposables in number fields: A survey

Abstract

We give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discuss indecomposable algebraic integers as one of the main tools.
Paper Structure (8 sections, 85 equations)

This paper contains 8 sections, 85 equations.

Table of Contents

  1. Introduction
  2. The 15- and 290-Theorems
  3. Quadratic Lattices
  4. Real Quadratic Fields
  5. (Non-)Existence of Universal Forms
  6. General Results
  7. Families of Cubic Fields
  8. Lifting Problem for Universal Forms

Theorems & Definitions (8)

  • proof
  • proof
  • proof
  • proof
  • proof : Sketch of proof of Theorem $\ref{['theoremKT']}$
  • proof
  • proof : Sketch of proof.
  • proof : Sketch of proof of Theorem $\ref{['theoremKYLifting']}$