A Concrete Introduction to Higher Algebra

This is an introductory level textbook in number theory and higher algebra. I particularly liked the extremely clear language and style of the author. He explains most of the passages first in words and then in formulas, making all steps much less abstract than other algebra books tend to do. I would recommend this especially for self-study, as the book reads exactly as a good teacher talks to a class. Concerning the errata, this does not bother me. Go to the web site of the author (http://math.albany.edu/~lc802/), download the errata and in ten minutes you will have penciled in the corrections.

Let it be proclaimed as the first axiom of mathematics writing: Difficulty should never be multiplied without necessity. The informal prose is the one feature that stands out prominently throughout this book. The misty road towards a true understanding of the abstract and sometimes difficult concepts in modern algebra (group, ring, field, homomorphism etc) is paved with concrete examples and applications from number theory. This approach not only parallels the very historical development of the subject, but also has the pedagogical advantage that it does not divorce the theory from the practice. Laid to rest is the mythical belief that the whole edifice of modern algebra rests precariously on a theoretical foundation of purely axiomatic constructs separate from everyday reality: any student who can tell time on an analog clock or values the security of her credit card will appreciate the practical contribution of the field. My only complaint is that the book is rife with errors, of which some can be checked against the errata available on the author's website. Other than that, I lavish only praise upon it.

The text is one of the finest I've seen; However, there are not enough excercises, and there is no warning as to whether the problems relate to the text or are problems for further study. It's all right to have problems that are extensions of theory, but some of these problems are something else entirely.

This book is an introduction to abstract algebra. Essentially Professor Childs gives a minicourse on number theory as a precursor to abstract algebra. He then developes ring theory before group theory and takes a particularly historical approach to groups. There is a lot of serious attention to applications such as coding and cryptography (although both concern "codes" they are different areas). There is attention to finite fields and the foundations of Galois theory are given if not the full treatment. This book is a good text if not a great one, it has enough material to give the instructor flexibility in the course. It is a good book for the serious student to have in his/her library.