Abstract Witt Rings When Certain Binary Forms Represent Exactly Four Elements

Abstract

AbstractAn abstract Witt ring (R, G) of positive characteristic is known to be a group ring S[Δ] with ﹛1﹜ ≠ Δ ⊆ G if and only if it contains a form〈1,x〉, x ≠1, which represents only the two elements 1 and x. Carson and Marshall have characterized all Witt rings of characteristic 2 which contain binary forms representing exactly four elements. Such results which show R is isomorphic to a product of smaller rings are helpful in settling the conjecture that every finitely generated Witt ring is of elementary type. Here, some special situations are considered. In particular if char(R) = 8, |D〈l, 1〉| = 4, and R contains no rigid elements, then R is isomorphic to the Witt ring of the 2-adic numbers. If char(R) = 4, |D〈l,a〉| = 4 where a ∈ D〈1, 1〉, and R contains no rigid elements, then R is either a ring of order 8 or is the specified product of two Witt rings at least one of which is a group ring. In several cases R is realized by a field.