Algebraic Number Theory (Discrete Mathematics and Its Applications) 2nd Edition by Richard A. Mollin (PDF)

Description

Bringing the material up to date to reflect modern applications, Algebraic Number Theory, Second Edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws.New to the Second EditionReorganization of all chaptersMore complete and involved treatment of Galois theoryA study of binary quadratic forms and a comparison of the ideal and form class groupsMore comprehensive section on Pollard’s cubic factoring algorithmMore detailed explanations of proofs, with less reliance on exercises, to provide a sound understanding of challenging materialThe book includes mini-biographies of notable mathematicians, convenient cross-referencing, a comprehensive index, and numerous exercises. The appendices present an overview of all the concepts used in the main text, an overview of sequences and series, the Greek alphabet with English transliteration, and a table of Latin phrases and their English equivalents.Suitable for a one-semester course, this accessible, self-contained text offers broad, in-depth coverage of numerous applications. Readers are lead at a measured pace through the topics to enable a clear understanding of the pinnacles of algebraic number theory.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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.

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