Advanced Number Theory (Dover Books on Mathematics) by Harvey Cohn (PDF)

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A very stimulating book … in a class by itself. — American MathematicalMonthlyAdvanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker’s Basis Theorem for Abelian Groups.Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski’s theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints.In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory.

User’s Reviews

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

⭐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.

⭐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).

⭐The material is great and versatile. The main focus of this book is to present all theory relevant to the study of binary quadratic forms but includes an indtroduction to characters and the Dirilecht L-series as well. Although binary quadratic forms were completely solved by Gauss in 1800, the approach taken here is through theories that were crystalized and matured 50 or more years later. You can find this historical information scattered through the book.The empasis that the author tries to give that this book is mainly about quadratic forms and quadratic integers which allow one to solve them is very confusing. The book’s structure does not reflect that. Quadratic forms are more like an application that gives “practical relevance” to everything else. This book gives a lot of prominence and empasis on advanced concepts while giving little thought on pedagogical principles so that the reader has an easier task of navigating himself through this very demanding subject.For example, after characterizing the quadratic integers and demonstrating their relationship to quadratic forms, it proceeds with a chapter on lattices. A discussion of units and the failure of unique factorization comes after that, not depending on any material from the chapter on lattices. The title of that chapter is the impressive “Basis theorems”, using the innocent looking concept of module basis while the chapter is about the advanced concept of the lattice.For someone already familiar with algebraic number theory this book may be an interesting read because otherwise the presentation is certainly not dry or dull. The peculiar organization of the material and the imaginative titles cerainly help with that. A first-time reader on the other hand is going to have a very hard time with this book, especiall if doing self-study.

⭐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.

⭐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.

⭐ok

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