PREFACE :
This book on algebraic systems is designed to be used either as a supplement to current texts or
as a stand-alone text for a course in modern abstract algebra at the junior and/or senior levels.
In addition, graduate students can use this book as a source for review. As such, this book is
intended to provide a solid foundation for future study of a variety of systems rather than to be
a study in depth of any one or more.
The basic ingredients of algebraic systems–sets of elements, relations, operations, and mappings–are discussed in the first two chapters. The format established for this book is as follows:
The basic ingredients of algebraic systems–sets of elements, relations, operations, and mappings–are discussed in the first two chapters. The format established for this book is as follows:
. a simple and concise presentation of each topic
. a wide variety of familiar examples
. proofs of most theorems included among the solved problems
. a carefully selected set of supplementary exercises
In this upgrade, the text has made an effort to use standard notations for the set of natural numbers, the set of integers, the set of rational numbers, and the set of real numbers. In addition, definitions are highlighted rather than being embedded in the prose of the text.
Also, a new chapter (Chapter 10) has been added to the text. It gives a very brief discussion of Sylow Theorems and the Galois group.
The text starts with the Peano postulates for the natural numbers in Chapter 3, with the various number systems of elementary algebra being constructed and their salient properties discussed. This not only introduces the reader to a detailed and rigorous development of these number systems but also provides the reader with much needed practice for the reasoning behind the properties of the abstract systems which follow.
. a wide variety of familiar examples
. proofs of most theorems included among the solved problems
. a carefully selected set of supplementary exercises
In this upgrade, the text has made an effort to use standard notations for the set of natural numbers, the set of integers, the set of rational numbers, and the set of real numbers. In addition, definitions are highlighted rather than being embedded in the prose of the text.
Also, a new chapter (Chapter 10) has been added to the text. It gives a very brief discussion of Sylow Theorems and the Galois group.
The text starts with the Peano postulates for the natural numbers in Chapter 3, with the various number systems of elementary algebra being constructed and their salient properties discussed. This not only introduces the reader to a detailed and rigorous development of these number systems but also provides the reader with much needed practice for the reasoning behind the properties of the abstract systems which follow.
Chapter 16 as an example of a non-commutative polynomial ring. The characteristic
polynomial of a square matrix over a field is then defined. The characteristic roots
and associated invariant vectors of real symmetric matrices are used to reduce the equations
of conics and quadric surfaces to standard form. Linear algebras are formally defined in
Chapter 17 and other examples briefly considered.
In the final chapter (Chapter 18), Boolean algebras are introduced and important applications to simple electric circuits are discussed.
The co-author wishes to thank the staff of the Schaum’s Outlines group, especially Barbara Gilson, Maureen Walker, and Andrew Litell, for all their support. In addition, the co-author wishes to thank the estate of Dr. Frank Ayres, Jr. for allowing me to help upgrade the original text.
In the final chapter (Chapter 18), Boolean algebras are introduced and important applications to simple electric circuits are discussed.
The co-author wishes to thank the staff of the Schaum’s Outlines group, especially Barbara Gilson, Maureen Walker, and Andrew Litell, for all their support. In addition, the co-author wishes to thank the estate of Dr. Frank Ayres, Jr. for allowing me to help upgrade the original text.
Title : Schaums outline of theory and problems of abstract algebra
author(s) : Lloyd Jaisingh, Frank Ayres
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