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## Über dieses Buch

From the reviews: "O'Meara treats his subject from this point of view (of the interaction with algebraic groups). He does not attempt an encyclopedic coverage ...nor does he strive to take the reader to the frontiers of knowledge... . Instead he has given a clear account from first principles and his book is a useful introduction to the modern viewpoint and literature. In fact it presupposes only undergraduate algebra (up to Galois theory inclusive)... The book is lucidly written and can be warmly recommended.
J.W.S. Cassels, The Mathematical Gazette, 1965
"Anyone who has heard O'Meara lecture will recognize in every page of this book the crispness and lucidity of the author's style;... The organization and selection of material is superb... deserves high praise as an excellent example of that too-rare type of mathematical exposition combining conciseness with clarity...
R. Jacobowitz, Bulletin of the AMS, 1965

## Inhaltsverzeichnis

### Chapter I. Valuated Fields

Abstract
The descriptive language of general topology is known to all mathematicians. The concept of a valuation allows one to introduce this language into the theory of algebraic numbers in a natural and fruitful way. We therefore propose to study some of the connections between valuation theory, algebraic number theory, and topology. Strictly speaking the topological considerations are just of a conceptual nature and in fact only the most elementary results on metric spaces and topological groups will be used; nevertheless these considerations are essential to the point of view taken throughout this chapter and indeed throughout the entire book.
O. Timothy O’Meara

### Chapter II. Dedekind Theory of Ideals

Abstract
In Chapter I we studied the ring of integers o (p) of a single non- archimedean spot p. We shall see in §33J that the set of algebraic integers of a number field F can be expressed in the form
$$o(S) = \mathop \cap \limits_{p \in S} o(p)$$
where S consists of all non-archimedean spots on F. This exhibits a strong connection between the algebraic integers and the prime spots of a number field, and we shall start to exploit it here. Specifically, we shall use the theory of prime spots to set up an ideal theory in o (S). For the present we can be quite general and we consider an arbitrary field F that is provided with a set of spots satisfying certain axioms. We shall call these axioms the Dedekind axioms for S since they lead to Dedekind’s ideal theory in o (S).
O. Timothy O’Meara

### Chapter III. Fields of Number Theory

Abstract
The first two chapters have been done in great generality. Before we can move on to the deeper results of number theory we shall have to make additional assumptions about the underlying field F. We shall do this by explicitly stating our fields of interest. They are the field of rational numbers or any field of rational functions in one variable over a finite field of coefficients, all finite extensions of these fields, and all completions thereof. By restricting ourselves to these fields we obtain two additional properties. Roughly speaking, the first of these properties is one of finiteness of the residue class field and the second is one of dependence among the valuations. These are actually the decisive properties that distinguish the rest of the arithmetic theory from the first two chapters. In fact it is possible to axiomatize these properties1 and to show that they lead directly to the fields of number theory, but we shall not go into that here.
O. Timothy O’Meara

### Chapter IV. Quadratic Forms and the Orthogonal Group

Abstract
We leave the arithmetic theory of fields in order to develop a different subject, the abstract theory of quadratic forms. In the latter half of the book we shall combine these two subjects into the arithmetic theory of quadratic forms. Our immediate purpose is to introduce a quadratic form and an orthogonal geometry on an arbitrary finite dimensional vector space and to study certain groups of linear transformations that leave the quadratic form invariant. We must make the assumption from now on that the field of scalars F does not have characteristic 2. As we indicated, our vector spaces are assumed to be finite dimensional.
O. Timothy O’Meara

### Chapter V. The Algebras of Quadratic Forms

Abstract
Our purpose in this chapter is to introduce three algebras of importance in the theory of quadratic forms, the Clifford algebra, the quaternion algebra, and the Hasse algebra. The Clifford algebra will be developed from first principles and its main use for us will be in the definition of an invariant called the spinor norm. The quaternion algebra and the Hasse algebra play an important role in the arithmetic theory of quadratic forms. The definition of the Hasse algebra depends on some of the structure theory of central simple algebras, in particular it needs Wedderburn’s theorem and the theory of similarity of algebras that is normally used in defining the Brauer group. We have therefore included a proof of Wedderburn’s theorem and some of its consequences. Also included as a convenience to the reader is a brief discussion of the tensor product of finite dimensional vector spaces1.
O. Timothy O’Meara

### Chapter VI. The Equivalence of Quadratic Forms

Abstract
One of the major accomplishments in the theory of quadratic forms is the classification of the equivalence class of a quadratic form over arithmetic fields. We are ready to present this part of the theory. Roughly speaking it goes as follows: the global solution is completely described by local and archimedean solutions, the local solution involves the dimension, the discriminant, and an invariant called the Hasse symbol, the complex archimedean solution is trivial, and the real archimedean solution is the well-known law of inertia of Sylvester.
O. Timothy O’Meara

### Chapter VII. Hilbert’s Reciprocity Law

Abstract
The Hilbert Reciprocity Law states that
$$\prod\limits_p {\left( {\frac{{\alpha ,\beta }}{p}} \right)} = 1.$$
The major portion of this chapter is devoted to the proof of this formula for algebraic number fields. The formula is actually true over any global field, but we shall not go into the function theoretic case here. The Hilbert Reciprocity Law gives a reciprocity law for Hasse symbols, namely
$$\prod\limits_p {{S_p}V} = 1,$$
and this can be regarded as a dependence relation among the invariants of the quadratic space V. We shall investigate the full extent of this dependence in § 72.
O. Timothy O’Meara

### Chapter VIII. Quadratic Forms over Dedekind Domains

Abstract
The rest of the book is devoted to a study of the equivalence of quadratic forms over the integers of local and global fields. Our first purpose in the present chapter is to state the nature of this problem in modern terminology and in the general setting of an arbitrary Dedekind domain. Our second purpose is to develop some technique in this general situation. The more interesting results must wait until we specialize to the fields of number theory.
O. Timothy O’Meara

### Chapter IX. Integral Theory of Quadratic Forms over Local Fields

Abstract
This chapter classifies quadratic forms under integral equivalence over local fields1.
O. Timothy O’Meara

### Chapter X. Integral Theory of Quadratic Forms over Global Fields

Abstract
We conclude this book by introducing the genus and the spinor genus of a lattice on a quadratic space over a global field, and by studying the relation between these two new objects and the class. We shall use these relations to obtain sufficient conditions under which two lattices are in the same class.
O. Timothy O’Meara

### Backmatter

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