Levels in algebra and topology

Abstract

The level s(A) of a (commutative) ring A is the smallest natural number s such that -1 is a sum of s squares in A. [l,p. 184]).In this note, we announce the following. While the above results are of an algebraic nature, their proofs (at least as so far discovered) are purely topological. One uses ideas from homotopy and

Levels of quaternion algebras

“…2. If −1 is a sum of squares in R, then s(R) = min{n | ∃x 1 , · · · , x n ∈ R : −1 = x 2 1 + · · · + x 2 n } In the early 1930s, Van der Waerden asked which values can arise as level of a field. At the time, all fields where the level was known and finite had level 1, 2 or 4.…”

“…The level question for integral domains was solved in 1980 by Z. D. Dai, T. Y. Lam and C. K. Peng [2] who proved that any positive integer can occur as level of an integral domain, more precisely, they showed that the integral domain R = R[X 1 , · · · , X n ]/(1 + X 2 1 + · · · + X 2 n ) has level n. The proof is topological in nature and invokes the Borsuk-Ulam theorem. Incidentally, the quotient field F = Quot(R) has level 2 k where k is such that 2 k ≤ n < 2 k+1 , yielding Pfister's original examples of fields whose level is a prescribed 2-power.…”

On the level of principal ideal domains

“…Related to the level s(A) of a ring A, there is another invariant, called the sublevel σ(A) of A (see [4]). It is the smallest natural number r such that there is a unimodular vector (a 1 , .…”

“…On the other hand, in [5] (see also [4]) it is shown that any natural number can be realized as the level of some ring A.…”

An algebraic proof of the Borsuk-Ulam theorem for polynomial mappings

“…Lam has shown me the proofs of many other algebraic results on sums of squares and higher powers, partially announced in [1], which he and his collaborators gained by topological methods. Up to now I know of no way to prove any of these results algebraically.…”