\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2015d.00714}
\itemau{Dixon, Martyn R.; Kurdachenko, Leonid A.; Subbotin, Igor Ya.}
\itemti{An introduction to essential algebraic structures.}
\itemso{Hoboken, NJ: John Wiley \& Sons (ISBN 978-1-118-45982-9/hbk; 978-1-118-49776-0/ebook). viii, 232~p. (2015).}
\itemab
Abstract algebra is an important part of mathematics program at any university. Today, abstract algebra is the basis of much of information technology and accompanying need for computer security and is also of interest to physicist, chemists and other scientist. There are even applications of abstract algebra to music theory. This book was written in order to be appropriate for typical students in computer science, mathematics, mathematics education and other disciplines, but the authors expect the book to be of interest not only to mathematics major, but also to anyone who would like to learn the basic topics of modern algebra. The book is structured into five chapters. The first chapter, Sets, investigates the set, functions and matrix theory, three fundamental concepts in mathematics. In Chapter 2, Numbers, we can find the main properties of integers, rational, irrational and real numbers, viewed from an algebraic point of view. The next three chapters introduce the fundamental algebraic structures: groups, rings and fields. Groups are the first algebraic structures that are presented. Their study was motivated by the old problem of founding a formula for the roots of a polynomial in terms of the coefficients of that polynomial. We can find in this chapter aspects regarding groups, subgroups, cosets and normal subgroups, factor groups and homomorphisms. Investigating the sets of numbers, of polynomials, of real and complex functions, the set of matrices or the set of vectors, ``it was realized many years ago that these sets exhibit many common features and these properties form a basis of the definition of a ring." In Chapter 4, Rings, the authors presents the main properties of rings, subrings, associative rings, rings of polynomials, ideals and quotient rings or homomorphisms of rings. Currently, field theory is one of the most advanced algebraic theory, having a variety of connections with other areas of mathematics. Besides the basic properties, in Chapter 5 we can find field extensions and fields of algebraic numbers. Numerous examples are presented throughout every chapter and an exercises set after each chapter section ``in an effort to built a deeper understanding of the subject and improve knowledge retention". Hints and answers of these exercises are presented in the end of the book.
\itemrv{Florentina Chirte\c{s} (Craiova)}
\itemcc{H45}
\itemut{sets; numbers; groups; rings; fields}
\itemli{}
\end