Polynomials Annihilating the Witt Ring

Let F be a non-formally real field of characteristic not 2 and let W(F) be the Witt ring of F. In certain cases generators for the annihilator ideal are determined. Aim the primary decomposition of A(F) is given. For formally d fields F, as an analogue the primary decomposition of A_t(F) = {f(X) ∈ Z[X]| f(ω) = 0 for all ω ∈ W_t(F)}, where W_t(F) is the torsion part of the Witt group, is obtained.