Let R be a ring with 1 ≠ 0. The level s(R) of R is the least integer n such that -1 is a sum of n squares in R provided such an integer exists, otherwise one defines the level to be infinite. In this survey, we give an overview on the history and the major results concerning the level of rings and some related questions on sums of squares in rings with finite level. The main focus will be on levels of fields, of simple noncommutative rings, in particular division rings, and of arbitrary commutative...

Let $R$ be a 2-dimensional normal excellent henselian local domain in which $2$ is invertible and let $L$ and $k$ be its fraction field and residue field respectively. Let ${\Omega }_R$ be the set of rank 1 discrete valuations of $L$ corresponding to codimension 1 points of regular proper models of $SpecR$. We prove that a quadratic form $q$ over $L$ satisfies the local-global principle with respect to ${\Omega }_R$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ and $R=A\left[\right[y\left]\right]$, where $A$ is a complete discrete valuation ring with...