If ϕ is an excellent form, then it is possible to use the dimensions of the higher complements of ϕ to obtain an annihilating polynomial of ϕ of low degree. The main result of this paper is the construction of such a polynomial with the help of methods from the theory of generic splitting of quadratic forms. Mathematics Subject Classification (2000). Primary 11E81; Secondary 16R99.

We study annihilating polynomials and annihilating ideals for elements of Witt rings for groups of exponent 2. With the help of these results and certain calculations involving the Clifford invariant we are able to give full sets of generators for the annihilating ideal of both the isometry class and the equivalence class of an arbitrary quadratic form over a local field. By applying the Hasse-Minkowski theorem we can then achieve the same for an arbitrary quadratic form over a global field.

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