Advanced Number Theory
"A very stimulating book ... in a class by itself." — American Mathematical Monthly
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In a nutshell, the book serves as an introduction to Gauss' theory of quadratic forms and their composition laws (the cornerstone of his Disquisitiones Arithmeticae) from the modern point of view (ideals in quadratic number fields). I strongly recommend it as a gentle introduction to algebraic number theory (with exclusive emphasis on quadratic number fields and binary quadratic forms). As a bonus, the book includes material on Dirichlet L-functions as well as proofs of Dirichlet's class number formula and Dirichlet's theorem in primes in arithmetic progressions (of course this material requires the reader to have the background of a one-semester course in real analysis; on the other hand, this material is largely independent of the subsequent algebraic developments).
Better titles for this book would be "A Second Course in Number Theory" or "Introduction to quadratic forms and quadratic fields". It is not a very advanced book in the sense that required background is only a one-semester course in number theory. It does not assume prior familiarity with abstract algebra. While exercises are included, they are not particularly interesting or challenging (if probably adequate to keep the reader engaged).
While the exposition is *slightly* dated, it feels fresh enough and is particularly suitable for self-study (I'd be less likely to recommend the book as a formal textbook). Students with a background in abstract algebra might find the pace a bit slow, with a bit too much time spent on algebraic preliminaries (the entire Part I---about 90 pages); however, these preliminaries are essential to paving the road towards Parts II (ideal theory in quadratic fields) and III (applications of ideal theory).
It is almost inevitable to compare this book to Borevich-Shafarevich "Number Theory". The latter is a fantastic book which covers a large superset of the material in Cohn's book. Borevich-Shafarevich is, however, a much more demanding read and it is out of print. For gentle self-study (and perhaps as a preparation to later read Borevich-Shafarevich), Cohn's book is a fine read.
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Advanced Number Theory
Suggested reading:. T. Apostol, Introduction to Analytic Number Theory. H. Cohn, Advanced Number Theory; H. Davenport, Multiplicative Number Theory.
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11: Number theory
Cohn, Harvey: "Advanced number theory", Dover Publications, Inc., New York, 1980. 276 pp. ISBN 0-486-64023-X; Ireland, Kenneth F.; Rosen, Michael I.: "A ...
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H. Cohn, Advanced Number Theory. One of the few books with a readable account of quadratic forms. D. Flath, Introduction to number theory. ...
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NUMBER THEORY BOOKS, 1993 OR BEFORE
... pdta Elliot, Grundlehren der mathematischen Wissenschaften 239, Springer 1979; Advanced Number Theory (formerly A Second Course in Number Theory), ...
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Cohn H. Передовая теория числа (Дувр, 1980), Cohn H. Advanced ...
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