This thesis is a rather substantial recapitulation of the progress the theory of generic splitting of quadratic forms has made since it was first introduced in 1976 by Manfred Knebusch. The thesis deals with the results about this theory that can be obtained with elementary methods. Of special importance are the results by Detlev Hoffmann, since with the help of them it becomes possible to simplify the proofs of a numbers of less recent results.
Already Ernst Witt noticed, that the Witt ring (of quadratic forms) of a field is integral (over the integers), but it took until 1987 for David Lewis to construct specific polynomials (with integer coefficients), which annihilate the isometry and equivalence classes of quadratic forms of a given dimension over an arbitrary field. In this thesis we study annihilating polynomials for quadratic forms (i.e. for the isometry or equivalence class of a quadratic form). In particular we study the annihilating ideal for a given quadratic form, i.e. the ideal consisting of all annihilating polynomials for that quadratic form. It is our aim to achieve a general understanding of the structure of these annihilating ideals, and to develop methods that allow us to determine annihilating ideals for isometry and equivalence classes of quadratic forms over a given field.