Advanced Number Theory

Rating:
Summary: Not advanced enough for Fermat's last theorm
Review: For the money this book is a good buy. If you want to understand Wiles's use of modular forms in his proof of Fermat's last theorem, it isn't advanced enough! It also isn't an easy read, but it tries to cover the major areas. It is best in quadratic number theory and worst in Dirichlet L-series and Gaussian Sums, but it mentions just about every area of number theory. If you want "easy" ... look elsewhere.

Rating:
Summary: Not advanced enough for Fermat's last theorm
Review: For the money this book is a good buy. If you want to understand Wiles's use of modular forms in his proof of Fermat's last theorm, it isn't advanced enough! It also isn't an easy read, but it tries to cover the major areas. It is best in quadratic number theory and worst in Dirchlet L-series and Gaussian Sums, but it mentions just about every area of number theory. If you want "easy" as well as cheap, look elsewhere.

Rating:
Summary: All about quadratric number fields, but little else
Review: From 1962, this is a detailed account of quadratic number fields, and makes a fair introduction to the theory of number fields of any degree. Ideal theory (restricted to the quadratic case) is well covered in plenty of detail. Gauss's classic theory of binary quadratic forms is also included.

Cohn is clearly quite keen on the subject, and is not just writing a textbook on some arbitrary topic for which he thinks there might be a market. And he has no fear of including pedagogical remarks in a textbook. The English is a bit awkward in places, but that is a minor thing.

The basics about characters and Dirichlet L-series are developed, but only to the extent needed to give Dirichlet's original proof of his theorem on arithmetic progressions. That proof, unlike later ones, uses Dirichlet's class number formula for quadratic fields, and is worth a look.

There is a lengthy but now dated bibliography.

An unusual feature is a table (from Sommer's 1911 book) describing the structure of Z[sqrt(n)] for all nonsquare n from -99 to 99.

Rating:
Summary: A book only for advanced students !
Review: I had heard that this book would be difficult, but I wasn't prepared for Cohn's strange treatment of such important concepts as integral domains, modules, and ideals. He is pretty thorough, but I would have prefered a more structured development of the subject. All in all, the book was decent, but not something for a first-time learner of the subject to look to for guidance.

Rating:
Summary: nice text, but somewhat hard to read
Review: This book gives a nice treatment of the theory of quadratic number rings and fields. It also covers a few other topics like class field theory and the Dirichlet problem. It requires surprisingly little algebra, but this makes some parts of the text seem to drag if you already know the algebra he covers. Not really recommended for independent study, since some arguments are a bit confusing, and it has few examples.

Rating:
Summary: nice text, but somewhat hard to read
Review: This book gives a nice treatment of the theory of quadratic number rings and fields. It also covers a few other topics like class field theory and the Dirichlet problem. It requires surprisingly little algebra, but this makes some parts of the text seem to drag if you already know the algebra he covers. Not really recommended for independent study, since some arguments are a bit confusing, and it has few examples.

Rating:
Summary: Advanced, but now dated. Still useful.
Review: This is definitely an advanced book. But no book claiming to be advanced can hold that title for long since mathematical research is progressive. As advanced as the book is, it's just an introduction to advanced number theory now, and dated in places. This book was orginally published as "A Second Course in Number Theory " in 1962. I own several books by Harvey Cohn and I appreciate his writing style. He writes with the complete book in mind and every chapter and paragraph is cohesively developed. His writing (between the numerous equations, tables and proofs) is lucid and conversational with historical motivation. He places a strong emphasis on ideal theory, and quadratic fields. In this regard the book is almost redundant given his "Class Field Theory" book.

Be warned there is some dated material in this book. It is prior to Alan Baker's 1966 proof about d=-163 and imaginary quadratic fields, and such is still only conjectured in the text. And of course, FLT wasn't on Wiles' check list when this book was published.

It doesn't cover prime-producing polynomials or transcendental functions and their relation to class field theory, like one would hope (I guess the world had to wait for Baker for that). And forget about rational points on elliptic curves, none at all. It's from the period when elliptic equations were poo-pooed as relics before being brought to the fore again by recent developments.

Despite all the short-comings, I can still recommend the book as a worthy edition to your number theory library. Just don't put it at the top of your lists (unless you're short on cash and Dover is all you can afford).

Rating:
Summary: Advanced, but now dated. Still useful.
Review: This is definitely an advanced book. But no book claiming to be advanced can hold that title for long since mathematical research is progressive. As advanced as the book is, it's just an introduction to advanced number theory now, and dated in places. This book was orginally published as "A Second Course in Number Theory " in 1962. I own several books by Harvey Cohn and I appreciate his writing style. He writes with the complete book in mind and every chapter and paragraph is cohesively developed. His writing (between the numerous equations, tables and proofs) is lucid and conversational with historical motivation. He places a strong emphasis on ideal theory, and quadratic fields. In this regard the book is almost redundant given his "Class Field Theory" book.

Be warned there is some dated material in this book. It is prior to Alan Baker's 1966 proof about d=-163 and imaginary quadratic fields, and such is still only conjectured in the text. And of course, FLT wasn't on Wiles' check list when this book was published.

It doesn't cover prime-producing polynomials or transcendental functions and their relation to class field theory, like one would hope (I guess the world had to wait for Baker for that). And forget about rational points on elliptic curves, none at all. It's from the period when elliptic equations were poo-pooed as relics before being brought to the fore again by recent developments.

Despite all the short-comings, I can still recommend the book as a worthy edition to your number theory library. Just don't put it at the top of your lists (unless you're short on cash and Dover is all you can afford).