On the level. (English) Zbl 0677.12009
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.
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.
Reviewer: M.Kula
MSC:
12D15 | Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) |
11E04 | Quadratic forms over general fields |
11P05 | Waring’s problem and variants |