Vol. 109, No. 1, 1983

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Witt kernels of function field extensions

Robert Fitzgerald

Vol. 109 (1983), No. 1, 89–106

Let F be a field of characteristic not 2. For a non-hyperbolic quadratic form q of dimension at least 2, let F(q) denote the function field of the projective variety q = 0. We consider the problem, explicitly raised as problem D by Lam, of determining the kernel of induced map of Witt rings WF WF(q). This kernel is the Witt kernel of the field extension and is denoted by W(F(q)∕F). The basic tool is a comparison of W(F(q ⊥⟨x)∕F) and W(F(q)∕F). The Witt kernels W(F(q)∕F) where q has small dimension or F has small Hasse number are determined. Applications are made to the question of when a conservative form is embeddable.

Mathematical Subject Classification 2000
Primary: 11E81
Secondary: 18F25
Received: 28 January 1982
Published: 1 November 1983
Robert Fitzgerald