Matej Brešar

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The term algebra is used in different contexts. We learn elementary algebra in high school, and linear algebra in one of the first mathematical courses at university. This book provides an introduction to the algebra that is more precisely called abstract algebra. And it really is abstract. We are able to illustrate abstract algebraic concepts through concrete examples and soon get used to handling them. But what is their purpose? Questions like this often distract beginners. The basic definitions are so general that it can be difficult to imagine how they can be useful. And yet, it is a fact that algebraic methods are an extremely effective tool for solving various kinds of mathematical problems. Before we can begin to use them, however, we need to truly understand them—and this takes time. Some examples of their applicability will be presented in this book, but only in the last chapters. We comprehend algebra when we grasp the whole picture: the introduction of abstract concepts and their elementary consideration gradually lead to solutions of difficult concrete problems. The proofs are often short and elegant. There is not much computing; instead, intertwining seemingly simple ideas is what yields results. Algebra is stunningly beautiful. The book is based on lectures I gave to second year undergraduate students at the University of Ljubljana during the last four academic years. The original version is written in Slovenian. The present English version is not only a translation, but contains additional material and some modifications. There is a myriad of books on basic abstract algebra, so why does the world need another one? The present book covers the usual topics one would expect to find. Its special feature, however, is that these topics are grouped together in a non-traditional way, thereby giving the reader a somewhat different perspective on the main concepts of algebra than one gets from reading standard textbooks. ……