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**About this book :- **
**Schaum’s Outline Theory and Problems (2E) ** written by
** Frank Ayres **, Jr., Ph.D. and
** Lloyed R. Jaisingh**, Professor of Mathematics, Morehead State University
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:
. 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.
The first abstract algebraic system – the Group – is considered in Chapter 9. Cosets of a subgroup, invariant subgroups, and their quotient groups are investigated as well. Chapter 9 ends wi th the Jordan–Ho¨ lder Theorem for finite groups. Rings, Integral Domains Division Rings, Fields are discussed in Chapters 11–12 while Polynomials over rings and fields are then considered in Chapter 13. Throughout these chapters, considerable attention is given to finite rings. Vector spaces are introduced in Chapter 14. The algebra of linear transformations on a vector space of finite dimension leads naturally to the algebra of matrices (Chapter 15).
Matrices are then used to solve systems of linear equations and, thus provide simpler solutions to a number of problems connected to vector spaces. Matrix polynomials are discussed in 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 brie fly 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.

**Book Detail :- **
** Title: ** Schaum’s Outline Theory and Problems
** Edition: ** Second Edition
** Author(s): ** Frank Ayres, Lloyed Jaisingh
** Publisher: ** McGraw-Hill
** Series: **
** Year: ** 2003
** Pages: **
** Type: ** PDF
** Language: ** Englsih
** ISBN: ** 9780071403276,0071403272
** Country: ** US

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**About Author :- **

The author Frank Ayres, Jr. was a mathematics professor, best known as a famous author for the popular Schaum's Outlines series.

Ayres earned his bachelor of science degree from Washington College, Maryland and his master's and doctoral degrees from the University of Chicago.

He taught during 1921–4 at Ogden College and another four years at Texas A&M before coming to Dickinson College in 1928. He was promoted to associate professor in June, 1935. In 1943 he was named the Susan Powers Hoffman Professor of Mathematics. From 1938 until his retirement in June, 1958, he served as chairman of the mathematics department. Ayres was also an instructor in the Army Air Corps program at the College, 1943–44, and authored Basic Mathematics of Aviation, he wrote seven textbooks.

**Other famous books of similar Author :- **

Here is list all books avaliable of this author.

** • Download PDF Schaum's Outlines Calculus (5E) by Frank Ayres, Elliott Mendelson ****
• Download PDF Schaum’s Outline Theory and Problems of Abstract Algebra (2E) by Frank Ayres, Lloyd Jaisingh
• Download PDF Schaum Differential and Integral Calculus (3E) by Frank Ayres, Elliott Mendelson
• Download PDF Schaum's Easy Outline Calculus Crash Course by Frank Ayres, Elliott Mendelson
• Download PDF Schaum's 3000 Solved Problems in Calculus by Elliot Mendelson
• Download PDF Schaum's Outline of Theory & Problems of Beginning Calculus (2E) by Elliott Mendelson
**

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**Book Contents :- **
**Schaum’s Outline Theory and Problems (2E) ** written by
** Frank Ayres, Lloyed Jaisingh **.
cover the following topics.

**Part-1 Sets and Relations**

1. Sets

2. Relations and Operations
**Part-2 Number Systems
3. The Natural Numbers
4. The Integers
5. Some Properties of Integers
6. The Rational Numbers
7. The Real Numbers
8. The Complex Numbers
9. Groups
10. Further Topics on Group Theory
11. Rings
12. Integral Domains, Division Rings, Fields
13. Polynomials
14. Vector Spaces
15. Matrices
16. Matrix Polynomials
17. Linear Algebras
18. Boolean Algebras
**

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