Basic Algebra I (Dover Books on Mathematics) - Softcover

Jacobson, Nathan

Softcover

ISBN 10: 0486471896 ISBN 13: 9780486471891

Verlag: Dover Publications Inc., 2009

Inhaltsangabe

A classic text and standard reference for a generation, this volume and its companion are the work of an expert algebraist who taught at Yale for two decades. Nathan Jacobson's books possess a conceptual and theoretical orientation, and in addition to their value as classroom texts, they serve as valuable references.
Volume I explores all of the topics typically covered in undergraduate courses, including the rudiments of set theory, group theory, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. Its comprehensive treatment extends to such rigorous topics as Lie and Jordan algebras, lattices, and Boolean algebras. Exercises appear throughout the text, along with insightful, carefully explained proofs. Volume II comprises all subjects customary to a first-year graduate course in algebra, and it revisits many topics from Volume I with greater depth and sophistication.

Über diesen Titel

Über die Autorin bzw. den Autor

One of the world's leading researchers in abstract algebra, Nathan Jacobson (1910-95) taught at several prominent universities, including the University of Chicago, Johns Hopkins, and Yale.

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Basic Algebra I

By Nathan Jacobson

Dover Publications, Inc.

Copyright © 1985 Nathan Jacobson
All rights reserved.

ISBN: 978-0-486-47189-1

Contents

Preface, 
Preface to the First Edition, 
INTRODUCTION: CONCEPTS FROM SET THEORY. THE INTEGERS, 
1 MONOIDS AND GROUPS, 
2 RINGS, 
3 MODULES OVER A PRINCIPAL IDEAL DOMAIN, 
4 GALOIS THEORY OF EQUATIONS, 
5 REAL POLYNOMIAL EQUATIONS AND INEQUALITIES, 
6 METRIC VECTOR SPACES AND THE CLASSICAL GROUPS, 
7 ALGEBRAS OVER A FIELD, 
8 LATTICES AND BOOLEAN ALGEBRAS, 
Appendix, 
Index, 

CHAPTER 1

Monoids and Groups

The theory of groups is one of the oldest and richest branches of algebra. Groups of transformations play an important role in geometry, and, as we shall see in Chapter 4, finite groups are the basis of Galois' discoveries in the theory of equations. These two fields provided the original impetus for the development of the theory of groups, whose systematic study dates from the early part of the nineteenth century.

A more general concept than that of a group is that of a monoid. This is simply a set which is endowed with an associative binary composition and a unit—whereas groups are monoids all of whose elements have inverses relative to the unit. Although the theory of monoids is by no means as rich as that of groups, it has recently been found to have important "external" applications (notably to automata theory). We shall begin our discussion with the simpler and more general notion of a monoid, though our main target is the theory of groups. It is hoped that the preliminary study of monoids will clarify, by putting into a better perspective, some of the results on groups. Moreover, the results on monoids will be useful in the study of rings, which can be regarded as pairs of monoids having the same underlying set and satisfying some additional conditions (e.g., the distributive laws).

A substantial part of this chapter is foundational in nature. The reader will be confronted with a great many new concepts, and it may take some time to absorb them all. The point of view may appear rather abstract to the uninitiated. We have tried to overcome this difficulty by providing many examples and exercises whose purpose is to add concreteness to the theory. The axiomatic method, which we shall use throughout this book and, in particular, in this chapter, is very likely familiar to the reader: for example, in the axiomatic developments of Euclidean geometry and of the real number system. However, there is a striking difference between these earlier axiomatic theories and the ones we shall encounter. Whereas in the earlier theories the defining sets of axioms are categorical in the sense that there is essentially only one system satisfying them—this is far from true in the situations we shall consider. Our axiomatizations are intended to apply simultaneously to a large number of models, and, in fact, we almost never know the full range of their applicability. Nevertheless, it will generally be helpful to keep some examples in mind.

The principal systems we shall consider in this chapter are: monoids, monoids of transformations, groups, and groups of transformations. The relations among this quartet of concepts can be indicated by the following diagram:

[ILLUSTRATION OMITTED]

This is intended to indicate that the classes of groups and of monoids of transformations are contained in the class of monoids and the intersection of the first two classes is the class of groups of transformations. In addition to these concepts one has the fundamental concept of homomorphism which singles out the type of mappings that are natural to consider for our systems. We shall introduce first the more intuitive notion of an isomorphism.

At the end of the chapter we shall carry the discussion beyond the foundations in deriving the Sylow theorems for finite groups. Further results on finite groups will be given in Chapter 4 when we have need for them in connection with the theory of equations. Still later, in Chapter 6, we shall study the structure of some classical geometric groups (e.g., rotation groups).

1.1 MONOIDS OF TRANSFORMATIONS AND ABSTRACT MONOIDS

We have seen in section 0.2 that composition of maps of sets satisfies the associative law. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and βα is the map from S to U defined by (βα)(S) = β(α(s)) then we have γ(βα) = (γβ)α. We recall also that if 1T is the identity map t ->t on T, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and β1T = β for every α:S ->T and β: T ->U. Now let us specialize this and consider the set M(S) of transformations (or maps) of S into itself. For example, let S = {1, 2}. Here M(S) consists of the four transformations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where in each case we have indicated immediately below the element appearing in the first row its image under the map. It is easy to check that the following table gives the products in this M(S):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, generally, we have put ρσ in the intersection of the row headed by ρ and the column headed by σ (ρ, σ = 1, α, β, γ). More generally, if S = {1, 2, ..., n} then M(S) consists of nn transformations, and for a given n, we can write down a multiplication table like (1) for M(S). Now, for any non-vacuous S, M(S) is an example of a monoid, which is simply a non-vacuous set of elements, together with an associative binary composition and a unit, that is, an element 1 whose product in either order with any element is this element. More formally we give the following

DEFINITION 1.1.A monoid is a triple (M, p, 1) in which M is a non-vacuous set, p is an associative binary composition (or product) in M, and 1 is an element of M such that p( 1, a) = a = p(a, 1) for all a [member of] M.

If we drop the hypothesis that p is associative we obtain a system which is sometimes called a monad. On the other hand, if we drop the hypothesis on 1 and so have just a set together with an associative binary composition, then we obtain a semigroup (M, p). We shall now abbreviate p(a, b), the product under p of a and b, to the customary ab (or a · b). An element 1 of (M, p) such that a1 = a = 1a for all a in M is called a unit in (M, p). If 1' is another such element then 1'1 = 1 and 1'1 = 1', so 1' = 1. Hence if a unit exists it is unique, and so we may speak of the unit of (M, p). It is clear that a monoid can be defined also as a semi-group containing a unit. However, we prefer to stick to the definition which we gave first. Once we have introduced a monoid (M, p, 1), and it is clear what we have, then we can speak more briefly of "the monoid M," though, strictly speaking, this is the underlying set and is just one of the ingredients of (M, p, 1).

Examples of monoids abound in the mathematics that is already familiar to the reader. We give a few in the following list.

EXAMPLES

1. (N, +,0); N, the set of natural numbers, +, the usual addition...

Details zur Ausgabe

• Verlag: Dover Publications Inc.
• Erscheinungsdatum: 2009
• Sprache: Englisch
• ISBN 10: 0486471896
• ISBN 13: 9780486471891
• Einband: Taschenbuch
• Auflage: 2
• Anzahl der Seiten: 499
• Kontakt zum Hersteller: Dover; info@galileo-mc.de; ; Dachauer Str. 5-7; F�rstenfeldbruck; 82256; Deutschland