On \(p\)-quaternionic pairings of non-elementary type
This paper considers \(p\)-quaternionic pairings, first introduced by \textit{Y. S. Hwang} and \textit{B. Jacob} [Can. J. Math. 47, 527-543 (1995; Zbl 0857.12002)], which generalize to odd primes \(p\) the linked quaternionic mappings (\(p = 2\)) of \textit{M. Marshall} and \textit{J. Yucas} [Pac. J. Math. 95, 411-425 (1981; Zbl 0459.10013)], arising in the study of Witt rings of quadratic forms. The \(p\)-quaternionic pairings arising from local fields are said to be of local type. Hwang and Jacob have defined direct product and group extensions of \(p\)-quaternionic pairings, which are the appropriate analogues of the corresponding constructions in the theory of Witt rings. A finite \(p\)-quaternionic pairing is said to be of elementary type if it can be constructed from \(p\)-quaternionic pairings of local type via these two constructions. The main result of this paper is to show that for \(p \geq 3\), there exist \(p\)-quaternionic pairings of non-elementary type. This remains a significant open problem for \(p = 2\), i.e. the existence of finitely generated Witt rings of non-elementary type is still unknown. The method for finding these non-elementary-type pairings for \(p \geq 3\) is first to show that those of elementary type are not \`\` strongly regular\'\'\ unless they are either \`\` totally degenerated\'\'\ or of local type. The author then constructs pairings which are neither totally degenerated nor of local type.
finitely generated Witt rings of non-elementary type
scientific article; zbMATH DE number 1406356
scientific article