This is a superb book to introduce any math major to the most important ideas of abstract algebra. How often does a mathematician with the world renowned research stature of professor Artin choose to devote the time and effort to write a beginning book in essentially his specialty (he is a famous algebraic geometer), and really make the ideas understandable to the young student? Primarily when he has to teach the course, and wants a suitable source to teach from. That is apparently the genesis of this book, which took birth as class notes for the undergraduate algebra course at MIT taught several times by Professor Artin. This fine book can serve well as an introduction, or second course in algebra, at either honors or upper undergraduate level, or even first year graduate level, (although it omits one topic, multilinear and tensor algebra that grad students will eventually need). The material obviously rolls off the author's fingertips, and the reader is the beneficiary, since it all looks easy to us too, although one must definitely think along the way. With the easily defensible belief that the most important groups in mathematics are linear ones, the author begins with matrices, carries these illustrative examples throughout the treatment of groups, and even includes a beautiful introductory section on some important classical lie groups that cannot be found in any other comparable beginning algebra text. When an author is a true master, as here, he is not limited to mimicking the topics found in other successful texts but can cover each topic as he chooses and to the depth he wishes. Other unique touches are a section on divisibility in number fields, including Minkowski's geometric proof of the finiteness of the class group. This is the best modern introduction to abstract algebra available.

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