Vol. 51, No. 2, 1974

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Extended prime spots and quadratic forms

Ronald P. Brown

Vol. 51 (1974), No. 2, 379–395

Some of the local theory of extended prime spots on fields is developed here, with two applications in mind. In the first, two analogues to the Hasse-Minkowski theorem on equivalence of quadratic forms over global fields are developed, based on the notion of an ultracompletion of a field at an extended prime spot. They deal, respectively, with equivalence of quadratic forms over a simple transcendental extension of a global field, and with the reduced Witt ring of a general field. Examples illustrate problems involving the further extension of the global theory of quadratic forms. In the second application Harrison and Warner’s ultracompletions of a field at a finite or infinite prime are shown to be essentially ultracompletions at associated extended prime spots.

Mathematical Subject Classification 2000
Primary: 12J20
Secondary: 10C05
Received: 16 February 1973
Published: 1 April 1974
Ronald P. Brown