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Pfister Numbers Over Rigid Fields, Under Field Pfister Numbers over Rigid Fields, under Field Extensions and Related Concepts over Formally Real Fields Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) Der Fakult¨at fur¨ Mathematik der Technischen Universit¨at Dortmund vorgelegt von Nico Lorenz im Juni 2020 Dissertation Pfister Numbers over Rigid Fields, under Field Extensions and Related Concepts over Formally Real Fields Fakult¨atf¨urMathematik Technische Universit¨atDortmund Erstgutachter: Prof. Dr. Detlev Hoffmann Zweitgutachter: Prof. Thomas Unger, Ph.D. Tag der m¨undlichen Pr¨ufung:01.09.2020 Contents 1. Preface1 2. Important Results in the Theory of Quadratic Forms3 2.1. Basic Notation and General Facts......................3 2.2. Pfister Forms and Function Fields......................5 2.3. The Fundamental Ideal and Related Objects...............9 2.4. Quadratic Forms over Complete Discrete Valuation Fields....... 11 2.5. Field Extensions................................ 15 3. Structural Results for the Powers of IF 20 3.1. Known Pfister Numbers............................ 20 3.2. Auxiliary Results for Forms in InF ..................... 25 3.3. Upper Bounds for Forms with Good Subforms.............. 28 3.4. Pfister Numbers of Complete Discrete Valuation Fields......... 31 3.5. Forms in InF of Dimension 2n + 2n−1 .................... 34 3.A. Appendix: Another approach to forms of dimension 2n + 2n−1 in In .. 37 4. Rigid Fields 39 4.1. Introduction to the Theory of Rigid Fields................ 39 4.2. 14-dimensional I3-forms and 8-dimensional I2-forms.......... 47 4.3. 16-dimensional I3-forms............................ 49 4.4. Forms in InF of Dimension 2n + 2n−1 .................... 56 4.5. Asymptotic Pfister Numbers......................... 60 4.6. Symbol Length in InF ~In+1F ........................ 64 5. Pfister Numbers under Field Extensions 67 5.1. General Facts and Reduction Techniques................. 67 5.2. Quadratic Extensions............................. 69 5.3. Odd Degree Extensions............................ 72 5.4. Transcendental Extensions.......................... 73 6. Supreme Torsion Forms 79 6.1. Introduction................................... 79 6.2. 2-real-maximality................................ 83 6.3. Construction of Examples........................... 88 6.4. Correlations with Invariants......................... 92 7. Signature Ideals 94 7.1. Compatibility with In and X-decomposition............... 94 7.2. Transfer of Known Results.......................... 104 7.3. Counterexamples................................ 110 A. Foundations of Related Areas 113 A.1. Extensions of Valuations........................... 113 A.2. Orderings of Fields............................... 116 Bibliography 121 1. Preface It seems that any deeper investigation of the algebraic theory of quadratic forms requires a thorough knowledge of the theory of Pfister forms. W. Scharlau, Quadratic and Hermitian Forms, page 142. The above quotation pretty much sums up the main motivation for the upcoming thesis in just one sentence. The main goal for an algebraist working in the theory of quadratic forms is to understand the Witt ring of a field. This seems to be a really tough task, since even nowadays, approximately 90 years after the invention of the Witt ring, there are a lot of people researching in the beautiful area of quadratic forms to solve many different interesting problems concerning the Witt ring. As it seems to be pretty hard to study that Witt ring of a field in all its completeness, many researchers started to investigate the filtration of the Witt ring given by the ideals InF that are generated by the so-called Pfister forms. To understand the complexity of these ideals, a common way is to take a form ' lying in this ideal and to look for the minimal number of Pfister forms that are actually needed to represent the Witt class of '. This minimal number is called the Pfister number of ' and the main object in this thesis. For n ∈ {1; 2}, the problem to determine Pfister numbers is completely solved. For n = 3, we only have results for forms of small dimension. For forms of dimension 16 in I3F and higher, only some few partial results are known and even these require some heavy tools from related areas such as algebraic geometry, algebraic groups or even category theory. For n ≥ 4, only a few very special cases can be handled so far. It is thus convenient to first consider some easy cases rather than trying to solve the problem in all its generality. This is where this thesis starts. We will consider fields that are easy enough to develop special calculation techniques so that we can determine Pfister numbers of forms over such fields. Based on this research, some related problems occurred that are also approached. We now would like to give a quick overview of the thesis. The upcoming chapter is used to fix notation and recapturing standard results from the algebraic theory of quadratic forms that will be essential in the sequel. In the third chapter, we give a short introduction to the current state of research concerning the main problem and give some first further results that are obtained by adapting known techniques. Chapter 4 is completely devoted to so-called rigid fields. When restricting to this interesting class of fields, we are able to exactly determine the Pfister numbers for low-dimensional forms lying in an arbitrary high power of the fundamental ideal. We further provide upper bounds for the Pfister number for any form of higher dimension. 1 We change our point of view to ask a modified question in the next chapter. Given a field extension E~F and some form in InE, what can we say about its Pfister number if we presume knowing the behaviour of Pfister numbers over F ? In fact, for several important types of field extensions, including quadratic extensions, we can express the Pfister numbers over the bigger field in terms of Pfister numbers over the base field. The starting point for Chapter 6 is Karim Becher's article [Bec04]. Here, he used so called supreme Pfister forms to bound invariants that are related to Pfister numbers. Also with the upcoming chapter in mind, we adapted this concept to formally real fields. Even though we cannot deduce some further facts on the Pfister numbers itself, the developed theory leads to some examples of fields with interesting properties that reoccur in the last chapter. In Chapter 7, we extend the usual question to real fields while considering signature ideals and further generalizations of the fundamental ideal. To do so, we restrict ourselves to Pfister forms that are compatible with a given set of orderings. We try to generalize known results and provide examples in those cases where the results cannot be transferred to our new setting. Finally, in the appendix, we provide some important results from related theories that we use as tools in the main text. Primarily, these are results that are easy but technical consequences of well-known theorems that are rather non-standard. I would like to express my gratitude to my advisor Prof. Dr. Detlev Hoffmann for creating a pleasant working atmosphere and giving helpful advice whenever needed. Further I would like to thank my colleagues, especially Marco Sobiech, for helpful dialogues, both mathematically and non-mathematically. In addition I would like to thank Prof. Dr. Karim Becher for his interest in my work and inspiring discussions. Finally, I would like to thank my friends and my family and particularly my wife for supporting me on my way. 2 2. Important Results in the Theory of Quadratic Forms 2.1. Basic Notation and General Facts In this chapter, we will introduce some basic notation and record some fundamental results that will be used later. The notation is rather standard and compatible with at least one of the main sources [Lam05], [EKM08] or [Sch85]. Further, we will give some side facts to provide a wider framework for the reader's convenience. We assume all fields to be of characteristic not 2. By a quadratic form over F or just form for short, we always mean a finite dimensional non-degenerate quadratic form. Such forms can be diagonalized. As usual, we write ⟨a1; : : : ; an⟩ with suitable ∗ a1; : : : ; an ∈ F for such a diagonalization. Orthogonal sum and tensor product of quadratic forms are as usually denoted by ⊥ respective ⊗. The n-fold orthogonal sum of a quadratic form ' for n ∈ N is denoted by n × ' ∶= ' ⊥ ::: ⊥ '. The Witt ring of F , whose elements are in 1-1 correspondence to the isometry classes of anisotropic forms over F , is denoted by WF . By abuse of notation, we will often use the same symbol for a quadratic form, its anisotropic part and its Witt class in situations where confusion is unlikely. Similarly, dim(') will either mean the dimension of the underlying vector space or the value dim(') mod 2, depending on what we are working with. We will use the symbol ≅ to denote isometry of quadratic forms, while the equality sign = is used to indicate equality in the Witt ring. When we would like to emphasize that we are working in the Witt ring, we also use the usual symbols +; − and ⋅ for addition, subtraction and multiplication respectively rather than the symbols for the appropriate operation for the forms itself. As usual, if is a subform of ', we will write ⊆ '. Since the determinant of a quadratic form is not an invariant of its Witt class, we will sometimes need its ∗ ∗2 signed version denoted by d± ∶ WF → F ~F . We write H for the hyperbolic plane. The Witt index of ', i.e. the number of hyperbolic planes occurring in a Witt decomposition, will be denoted as iW ('). For the Witt index of an orthogonal sum of two forms, we have the following estimate. Proposition 2.1.1 ([Lam05, Chapter I. Exercise 16 (2)]): Let '1;'2 be (regular) quadratic forms over some field F . Then, for the orthogonal sum, we have iW ('1 ⊥ '2) ≤ iW ('1) + dim('2): By applying the above, we can easily deduce a criterion for a form to be a subform of a given quadratic form.
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