Annihilating Polynomials for Quadratic Forms

Let K be a field with char(K) ≠ 2. The Witt-Grothendieck ring (K) and the Witt ring W (K) of K are both quotients of the group ring ℤ[𝓖(K)], where 𝓖(K) := K*/(K*)2 is the square class group of K. Since ℤ[𝓖(K)] is integral, the same holds for (K) and W(K). The subject of this thesis is the study of annihilating polynomials for quadratic forms. More specifically, for a given quadratic form φ over K, we study polynomials P ∈ ℤ[X] such that P([φ]) = 0 or P({φ}) = 0. Here [φ] ∈ (K) denotes the isometry class and {φ} ∈ W(K) denotes the equivalence class of φ. The subset of ℤ[X] consisting of all annihilating polynomials for [φ], respectively {φ}, is an ideal, which we call the annihilating ideal of [φ], respectively {φ}. Chapter 1 is dedicated to the algebraic foundations for the study of annihilating polynomials for quadratic forms. First we study the general structure of ideals in ℤ[X], which later on allows us to efficiently determine complete sets of generators for annihilating ideals. Then we introduce a more natural setting for the study of annihilating polynomials for quadratic forms, i.e. we define Witt rings for groups of exponent 2. Both (K) and W(K) are Witt rings for the square class group 𝓖(K). Studying annihilating polynomials in this more general setting relieves us to a certain extent from having to distinguish between isometry and equivalence classes of quadratic forms. In Section 1.1 we study the structure of ideals in R[X], where R is a principal ideal domain. For an ideal I ⊂ R[X] there exist sets of generators, which can be obtained in a natural way by considering the leading coefficients of elements in I. These sets of generators are called convenient. By discarding super uous elements we obtain modest sets of generators, which under certain assumptions are minimal sets of generators for I. Let G be a group of exponent 2. In Section 1.2 we study annihilating polynomials for elements of ℤ[G]. With the help of the ring homomorphisms Hom(ℤ[G],ℤ) it is possible to completely classify annihilating polynomials for elements of ℤ[G]. Note that an annihilating polynomial for an element f ∈ ℤ[G] also annihilates the image of f in any quotient of ℤ[G]. In particular, Witt rings for G are quotients of ℤ[G]. In Section 1.3 we use the ring homomorphisms Hom(ℤ[G],ℤ) to describe the prime spectrum of ℤ[G]. The obtained results can then be employed for the characterisation of the prime spectrum of a Witt ring R for G. Section 1.4 is dedicated to proving the structure theorems for Witt rings. More precisely, we generalise the structure theorems for Witt rings of fields to the general setting of Witt rings for groups of exponent 2. Section 1.5 serves to summarise Chapter 1. If R is a Witt ring for G, then we use the structure theorems to determine, for an element x ∈ R, the specific shape of convenient and modest sets of generato