Orderings of higher level and semilocal rings
The concept of ordering of a field has been extended to that of a ``signature'' on a commutative semilocal ring \(A\), this being a homomorphism to the integers from the Witt ring of equivalence classes of nondegenerate symmetric bilinear forms over \(A\). This work has been carried out very successfully by M. Knebusch, A. Rosenberg and others. Recently E. Becker has developed a theory of ``higher level orderings'' for fields based on \(2^n\)-th powers rather than squares and Kleinstein and Rosenberg have defined corresponding Witt rings. In this paper these concepts are extended to semilocal rings. The main theorem states that if \(A\) is a local ring with residue field having more than two elements, then every ``signature of higher level'' has a corresponding prime ideal of \(A\) such that the signature extends uniquely to an ordering (of higher level) on the field of fractions of the quotient ring. Partial results are given for semilocal rings. From this theorem many of the usual results of Artin-Schreier theory can be extended to theorems concerning existence of these signatures and sums of \(2^n\)-th powers. For example, a local ring \(A\) has a signature of level \(n\) if and only if \(-1\) is not a sum of \(2^n\)-th powers. A corollary of this states the somewhat surprising result that if \(-1\) is a sum of squares, then it is a sum of \(2^n\)-th powers for every \(n\). Autorreferat.
scientific article; zbMATH DE number 3683744