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.

34 pages, journal version

References in corpus (4)

• Real quadratic fields with a universal quadratic form of given rank have density zero
• On continued fraction partial quotients of square roots of primes
• Periodicity of general multidimensional continued fractions using repetend matrix form
• Partitions into powers of an algebraic number

Cited by in corpus (6)

• Lifting problem for universal quadratic forms over totally real cubic number fields
• Universal quadratic forms and Northcott property of infinite number fields
• Most totally real fields do not have universal forms or Northcott property
• Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables
• Generalizing Hurwitz's quaternionic proof of Lagrange's and Jacobi's four-square theorems
• Euclidean lattices: theory and applications