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.
