Theory and problems of modern algebra Preface

This study of algebraic systems is designed to be used either as a supplement to current texts. or as a text for a course in modern abstract algebra at the junior-senior level. As such, it 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, mappings… are treated in the first two chapters. The pattern established here.

(i) a simple and concise presentation of each topic, (ii) a wide variety of familiar examples. (iii) proofs of most theorems included among the solved problems,

(iv) a carefully selected set of supplementary exercises, is followed throughout the book.

Beginning in Chapter 3 with the Peano postulates for the natural numbers. The several number systems of elementary algebra are constructed in turn and their salient properties deduced. This not only introduces the reader to a detailed and rigorous development of these number systems but also provides him with much needed practice for the deduction of 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,

and the chapter ends with the Jordan-Hölder Theorem for finite groups.

Chapters 10-11 are concerned with Rings. Integral Domains and Fields. Poly- nomials over rings and fields are then considered in Chapter 12 along with a certain amount of elementary theory of equations. Throughout these chapters, considerable attention is given to finite rings.

Vector spaces are introduced in Chapter 13. The algebra of linear transformations on a vector space of finite dimension leads naturally to the algebra of matrices (Chapter 14). Matrices are then used to solve systems of linear equations and, thus, provide simpler solutions of a number of problems connected with vector spaces. Matrix polynomials are considered in Chapter 15 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 sym- metric matrices are used to reduce the equations of conics and quadric surfaces to standard form. Linear algebras are formally defined in Chapter 16 and other examples briefly considered.

In the final chapter, Boolean algebras are introduced and the important applica- tion to simple electric circuits indicated.

The author wishes to take this opportunity to express his appreciation to the staff of the Schaum Publishing Company, especially to Jeffrey Albert and Alan Hopenwasser, for their unfailing cooperation.

Carlisle, Penna. June, 1965

FRANK AYRES, JR.