Let R be an arbitrary ring with identity element. The level s(R) is the smallest integer n such that -1 is a sum of n squares in R. Define \(s(R)=\infty\) if -1 is not a sum of squares in R. The paper under review is a short survey of various results concerning the level of fields, commutative and non-commutative rings. The connection with topology investigated by Z. D. Dai and T. Y. Lam in Comment. Math. Helv. 59, 376-424 (1984; Zbl 0546.10017) is presented, as well.
Most of the results are quoted without proofs. But the author describes the main idea of a proof of the important fact due to Dai, Lam and Peng, stating that every positive integer may occur as the level of a commutative ring. A brief outline of an algebraic proof of the Borsuk- Ulam theorem obtained by Arason and Pfister is also given.
The large list of references contains as many as 66 items.