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.
