Customer Reviews

Algebraic Number Theory (Discrete Mathematics and Its Applications)

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I learned a tremendous amount about Algebraic Number Theory from this excellent source. I have looked at other books that just skim the topics. This one covers them in depth and even has applications to cryptography (the author shuld have put that in the title). Even more advanced topics such as the higher reciprocity laws are covered with rigorous detail and extreme clarity. I read the AMS review for this book by Charles Parry and it is right on! This book should replace the old standards such as Janusz's and Marcus' books for instance. I'd say that this is a gem to be enjoyed.

I used this book in a one on one course in algebraic number theory in my fourth year of college. We finished just before ramification.

My background at the time included a year of undergraduate linear algebra, a year of undergraduate abstract algebra, a semester of intro. graduate algebra, intro. Galois theory, and intro. commutative algebra. The only things I used were Galois theory, my second abstract algebra course, and my second linear algebra course. Commutative algebra helped, but wasn't necessary in that abstract setting.

Organization: The book is very well organized with helpful appendices on abstract algebra basics (Groups, Rings, Fields) and Galois theory. The first chapter is slow-paced and provides a strong historical background for the material. A reviewer below suggested that there were "logical leaps" in the text--I never found such stuff, and I am always very picky about details. The author uses easy propositions that are assigned for homework sometimes, but they're mostly straightforward.

Exercises: They range from straightforward to quite thought-invoking... I remember one in particular, a starred problem, in which I had to use three "tricks" to solve.

Content: I like this book a lot. It's not super abstract on the level of Lang, but has hints of great generality throughout, and it's not some trivial algebraic number theory full of history, anecdotes, useless junk book with "Fermat's Last Theorem" misleadingly stated in the title somewhere. This book has a lot of stuff on applications to cryptography. The book covers things including

- Euclidean domains and unique factorization,

- special cases of FLT,

- Dirichlet's Unit Theorem,

- geometry of numbers,

- ideal class group,

- ramification,

- basics of class field theory,

- reciprocity laws.

It's a really nice all-around introduction. You need to be mature enough to read this book--the problems require that the reader is familiar with the relevent math. I was really impressed with the organization of this book--with topics like these, it's hard to have a nice balance of generality with concrete and useful results. This is for people not ready to appreciate Lang's book.

I am a graduate student specializing in Ring Theory and I have to tell you this is the absolute worst book I have ever had. Not only does the author make these humongous jumps in each section, he also has massive logical gaps. There are plenty of errors in the text starting right from the first section. You could easily spend a whole year deciphering (with a massive headache) the first chapter. The author is definetly wrong in assuming that all you need is a basic undergraduate number theory class and basic abstract algebra. You could have 2 comprehensive years of graduate modern algebra and still not be ready for the massive logical gaps in the book. Sure if we were all Galois and Eulers, the book would be easier, but I'd bet they'd even be scratching their heads often enough. My advice is stay away from the text at all cost. You'll regret paying the outrageous price for a text that is worth firepaper.