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000146786 0247_ $$2doi$$a10.5075/epfl-thesis-4683
000146786 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis4683-8
000146786 02471 $$2nebis$$a5985422
000146786 037__ $$aTHESIS_LIB
000146786 041__ $$aeng
000146786 088__ $$a4683
000146786 245__ $$aAnnihilating Polynomials for Quadratic Forms
000146786 269__ $$a2010
000146786 260__ $$aLausanne$$bEPFL$$c2010
000146786 300__ $$a121
000146786 336__ $$aTheses
000146786 520__ $$aLet 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 generators for the annihilating ideal of       x. In Chapter 2 we study annihilating polynomials for       quadratic forms over fields. More specifically, we first       consider fields K, over which quadratic forms can be       classified with the help of the classical invariants.       Calculations involving these invariants allow us to classify       annihilating ideals for isometry and equivalence classes of       quadratic forms over K. Then we apply methods from the       theory of generic splitting to study annihilating polynomials       for excellent quadratic forms. Throughout Chapter 2 we make       heavy usage of the results obtained in Chapter 1. Let K be a field with char(K) ≠ 2.       Section 2.1 constitutes an introduction to the algebraic       theory of quadratic forms over fields. We introduce the       Witt-Grothendieck ring (K) and the Witt ring       W(K), and we show that these are indeed Witt       rings for 𝓖(K). In addition we adapt the       structure theorems to the specific setting of quadratic       forms. In Section 2.2 we introduce Brauer groups and       quaternion algebras, and in Section 2.3 we define the first       three cohomological invariants of quadratic forms. In       particular we use quaternion algebras to define the Clifford       invariant. In Section 2.4 we begin our actual study of annihilating       polynomials for quadratic forms. Henceforth it becomes       necessary to distinguish between isometry and equivalence       classes of quadratic forms. We start by classifying       annihilating ideals for quadratic forms over fields K,       for which (K) and       W(K) have a particularly simple structure.       Subsequently we use calculations involving the first three       cohomological invariants to determine annihilating ideals for       quadratic forms over a field K such that       I3(K) = {0}, where       I(K) ⊂ W(K) is the       fundamental ideal. Local fields, which are a special class of       such fields, are studied in Section 2.5. By applying the       Hasse-Minkowski Theorem we can then determine annihilating       ideals of quadratic forms over global fields. Section 2.6 serves as an introduction to the elementary       theory of generic splitting. In particular we introduce       Pfister neighbours and excellent quadratic forms, which are       the subjects of study in Section 2.7. We use methods from       generic splitting to study annihilating polynomials for       Pfister neighbours. The obtained result can be applied       inductively to obtain annihilating polynomials for excellent       quadratic forms. We conclude the section by giving an       alternative, elementary approach to the study of annihilating       polynomials for excellent forms, which makes use of the fact       that (K) and       W(K) are quotients of       ℤ[𝓖(K)].
000146786 6531_ $$aquadratic forms
000146786 6531_ $$aannihilating polynomials
000146786 6531_ $$aintegral rings
000146786 6531_ $$aWitt rings
000146786 700__ $$aRühl, Klaas-Tido
000146786 720_2 $$0244699$$aBayer Fluckiger, Eva$$edir.$$g138858
000146786 8564_ $$s826725$$uhttps://infoscience.epfl.ch/record/146786/files/EPFL_TH4683.pdf$$yTexte intégral / Full text$$zTexte intégral / Full text
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000146786 918__ $$aSB$$cIMB$$dEDMA
000146786 919__ $$aCSAG
000146786 920__ $$b2010
000146786 970__ $$a4683/THESES
000146786 973__ $$aEPFL$$sPUBLISHED
000146786 980__ $$aTHESIS