Essential dimension of quadrics

Let X be an anisotropic projective quadric over a field F of characteristic not 2. The essential dimension dim_es(X) of X, as defined by Oleg Izhboldin, is dim_es(X)=dim(X)-i(X) +1, where i(X) is the first Witt index of X (i.e., the Witt index of X over its function field).Let Y be a complete (possibly singular) algebraic variety over F with all closed points of even degree and such that Y has a closed point of odd degree over F(X). Our main theorem states that dim_es(X)dim(Y) and that in the case dim_es(X)=dim(Y) the quadric X is isotropic over F(Y).Applying the main theorem to a projective quadric Y, we get a proof of Izhboldins conjecture stated as follows: if an anisotropic quadric Y becomes isotropic over F(X), then dim_es(X)dim_es(Y), and the equality holds if and only if X is isotropic over F(Y). We also solve Knebuschs problem by proving that the smallest transcendence degree of a generic splitting field of a quadric X is equal to dim_es(X). To the memory of Oleg Izhboldin