Universal quadratic forms and indecomposables in number fields: a survey (Q6594639)

The paper is an expository of results devoted to universal quadratic forms over the rings of integers in totally real number fields. The study of universal quadratic forms began from the famous 15-theorem of Conway and Schneeberger. Currently it turn to deep mathematical theory. The aim of the paper is to describe this theory (including recent developments) for beginners. Many considered results are obtained after 2015 and they are not discussed in previous expository papers. The author focuses on formulations of results, main ideas of proofs and some useful tools. Most of technical details are omitted.\N\NThe main topics are: constructions of universal quadratic forms; finiteness of the number of universal forms under some conditions on the field or on the rank of the form (including effective results); universal quadratic forms over the fields of small degrees (for example, for real quadratic fields) and indecomposable algebraic integers as the one of the main tools in study of universal quadratic forms.