Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

1985 | Buch

Basic Concepts Kapitel lesen Erstes Kapitel lesen

Quadratic and Hermitian Forms

verfasst von: Winfried Scharlau

Verlag: Springer Berlin Heidelberg

Buchreihe : Grundlehren der mathematischen Wissenschaften

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

For a long time - at least from Fermat to Minkowski - the theory of quadratic forms was a part of number theory. Much of the best work of the great number theorists of the eighteenth and nineteenth century was concerned with problems about quadratic forms. On the basis of their work, Minkowski, Siegel, Hasse, Eichler and many others crea­ ted the impressive "arithmetic" theory of quadratic forms, which has been the object of the well-known books by Bachmann (1898/1923), Eichler (1952), and O'Meara (1963). Parallel to this development the ideas of abstract algebra and abstract linear algebra introduced by Dedekind, Frobenius, E. Noether and Artin led to today's structural mathematics with its emphasis on classification problems and general structure theorems. On the basis of both - the number theory of quadratic forms and the ideas of modern algebra - Witt opened, in 1937, a new chapter in the theory of quadratic forms. His most fruitful idea was to consider not single "individual" quadratic forms but rather the entity of all forms over a fixed ground field and to construct from this an algebra­ ic object. This object - the Witt ring - then became the principal object of the entire theory. Thirty years later Pfister demonstrated the significance of this approach by his celebrated structure theorems.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Basic Concepts
Abstract
In this chapter we introduce the basic concepts which will occupy us in the first part of this book: quadratic forms and symmetric bilinear forms over fields of characteristic unequal to 2. The reader who already knows these basic concepts can immediately start with chapter 2. The assumption on the characteristic of the ground field is superfluous in places. However, its use will enable us to avoid a series of inconvenient case distinctions and special considerations for characteristic 2.
Winfried Scharlau
Chapter 2. Quadratic Forms over Fields
Abstract
The algebraic theory of quadratic forms originated in a classical paper by E. Witt [1937]. The importance of this paper consists of three essential contributions to the theory. First of all Witt introduced into the theory the geometrical language which is now commonly adopted. Secondly he constructed, in a canonical fashion, a commutative ring from the collection of all regular symmetric bilinear forms over a given field. This construction proved to be of fundamental importance because it allowed mathematicians to ask new and very fruitful questions: What is the structure of this ring and what does it tell us about the forms over the given field? Finally Witt summarized, unified, and extended the then known classification theorems. Despite the landmark importance of Witt’s paper, it was only after an incubation period of almost 30 years that a vigorous development of the algebraic theory of quadratic forms began. This new development started with the appearance of Pfister’s work in 1965 and 1966 which contains above all the first deep structure theorems about the Witt ring. The beauty and the elegance of these results led immediately to new questions, problems, and results and the theory has been flourishing ever since.
Winfried Scharlau
Chapter 3. Quadratic Forms over Formally Real Fields
Abstract
The very first theorem discovered in the algebraic theory of quadratic forms was the law of inertia of Jacobi and Sylvester. This theorem is concerned with quadratic forms over the field of real numbers. In the present chapter we will be interested in various generalizations of this result, and more generally in the connections between the theory of quadratic forms and the theory of ordered fields. Our ground field will be a formally real field, that is one in which — 1 cannot be expressed as a sum of squares. The theory of these fields was developed by Artin and Schreier in a series of now classical papers. Today it is a part of basic algebra. In the years since about 1970 it has been discovered that a substantial part of this theory can be developed in a simple and elegant manner in the framework of the theory of quadratic forms.
Winfried Scharlau
Chapter 4. Generic Methods and Pfister Forms
Abstract
It seems that any deeper investigation of the algebraic theory of quadratic forms requires a thorough knowledge of the theory of Pfister forms. They are the subject of this chapter. The investigation of Pfister forms is based essentially on the use of transcendental extensions of the ground field: One considers quadratic forms Σ a ij X i X j . with indeterminates X i . This approach leads naturally to the important concept of the function field of a quadratic form which is the function field of the quadratic Σ a ij X i X j =0.
Winfried Scharlau
Chapter 5. Rational Quadratic Forms
Abstract
Historically the theory of quadratic forms has its origins in number-theoretic questions of the following type: Which integers can be written in the form x 2 + 2y 2 , which are sums of three squares, or more generally, which integers can be represented by an arbitrary quadratic form Σ a ij x i x j integral coefficients? This general question is exceptionally difficult and we are still quite far from a complete solution. It is natural and considerably simpler to first investigate these questions over the field of rational numbers, that is, to ask for rational instead of integral solutions to the equation Σ a ij x i x j = a. This leads to the problem of classification of quadratic forms over \( \mathbb{Q} \), which was first solved by Minkowski. His solution appears in this chapter basically unaltered, except for a few simplifications and the use of modern terminology. The Gaussian sums of Gauss and Dirich-let play a significant role in the more formal algebraic part of the theory.
Winfried Scharlau
Chapter 6. Symmetric Bilinear Forms over Dedekind Rings and Global Fields
Abstract
Like the previous chapter this one deals with the number theoretic aspect of the theory of quadratic forms. Instead of the integers \( \mathbb{Z} \) and the rational field \( \mathbb{Q} \) we consider more generally an arbitrary algebraic number field and its ring of algebraic integers. Now and then we can be even more general by considering the quotient field of an arbitrary Dedekind domain. The theory of quadratic forms over algebraic number fields and the corresponding rings of integers is developed in detail in several books, especially O’Meara [1963] (to be referred to by OM). However, in the last few years several interesting results have been added to this theory. These new results, particularly those concerning the calculation of the Witt group are emphasized in this chapter.
Winfried Scharlau
Chapter 7. Foundations of the Theory of Hermitian Forms
Abstract
The definitions of symmetric bilinear forms and quadratic forms which appear in chapter 1 and are basic to the following chapters can be generalized. As already indicated in chapter 1 one can replace the ground field by an arbitrary commutative ring. In this case it is appropriate to define forms on finitely generated projective modules. However even the commutativity of the ground ring need not be assumed. Instead one can consider an associative ring with an involution and forms that are hermitian with respect to this involution. These hermitian forms are generalizations of symmetric bilinear forms. It turns out that generalizations of quadratic forms can also be defined in this context. This requires some technical preparation. But even this definition is still not general enough to encompass all concrete examples that we are interested in. For example, the forms on torsion modules considered in chapters 5 and 6 do not fit into this framework.
Winfried Scharlau
Chapter 8. Simple Algebras and Involutions
Abstract
The first part of this chapter contains basic results about finite-dimensional simple algebras due to Wedderburn, Dickson, Albert, R. Brauer, E. Noether, and others. This is a classical part of (linear) algebra. In many respects, it is similar to the algebraic theory of quadratic forms. In both cases, the theory is dominated by a few basic structure theorems. Subsequently, we develop Albert’s theory of involutions on simple algebras. There are many interesting connections between the theory of quadratic and hermitian forms on the one hand and the theory of simple algebras and involutions on the other. We want to mention the most important ones, though not all will be pursued in this book:
1)
With every quadratic form a simple algebra — its Clifford algebra — is associated in a functorial way. This algebra reflects many important properties of the quadratic form (see chapter 9).
 
2)
It will be shown in this chapter that the classification of involutions on simple algebras is almost identical with the classification of hermitian forms over division algebras.
 
3)
A particularly important part of the theory of hermitian forms is concerned with hermitian forms over group rings and group algebras. This theory is characterized by an attractive combination of methods from many parts of mathematics: representation theory, simple algebras, orders, quadratic forms, algebraic number theory, algebraic K-theory, and even algebraic topology.
 
And most basically:
4)
The very basis of the notion of hermitian forms is a ring with involution, and it is only natural that properties of this ring and involution are reflected in the pertinent theory of hermitian forms.
 
Winfried Scharlau
Chapter 9. Clifford Algebras
Abstract
With every quadratic space φ over a field K one can associate in a functorial way a K-algebra C(φ) called the Clifford algebra of φ. This construction is of fundamental importance in the algebraic theory of quadratic forms. In particular, the invariants e, d, c introduced in chapter 2 find a natural interpretation here. Many properties of these invariants are obvious in the context of Clifford algebras. Moreover, it is easy to include the case of characteristic 2.
Winfried Scharlau
Chapter 10. Hermitian Forms over Global Fields
Abstract
In this chapter we return to hermitian forms and continue the investigations begun in chapter 7 §1 and §6–9. As remarked in the introduction of that chapter these results correspond roughly to the basic results of chapter 1 about quadratic forms. Though the fundamentals of both theories — quadratic forms and hermitian forms — are quite similar, the two have grown in different directions. In particular, there does not exist an algebraic theory of hermitian forms that in content, extent, and importance is comparable to the algebraic theory of quadratic forms. One may speculate that in the future many problems from the algebraic theory of quadratic froms will be transferred to hermitian forms (and then perhaps will be transferred back using a suitable application of generic splitting fields and Frobenius functors). Nevertheless it seems that the theory of hermitian forms has a different character being more closely related to algebraic groups, algebras with involution, Galois cohomology, and algebraic K-theory.
Winfried Scharlau
Backmatter
Metadaten
Titel
Quadratic and Hermitian Forms
verfasst von
Winfried Scharlau
Copyright-Jahr
1985
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-69971-9
Print ISBN
978-3-642-69973-3
DOI
https://doi.org/10.1007/978-3-642-69971-9