The Theory of Algebraic Numbers (Dover Books on Mathematics) by Harry Pollard (PDF)

Description

Detailed proofs and clear-cut explanations provide an excellent introduction to the elementary components of classical algebraic number theory in this concise, well-written volume.The authors, a pair of noted mathematicians, start with a discussion of divisibility and proceed to examine Gaussian primes (their determination and role in Fermat’s theorem); polynomials over a field (including the Eisenstein irreducibility criterion); algebraic number fields; bases (finite extensions, conjugates and discriminants, and the cyclotomic field); and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture (concluding with discussions of Pythagorean triples, units in cyclotomic fields, and Kummer’s theorem). In addition to a helpful list of symbols and an index, a set of carefully chosen problems appears at the end of each chapter to reinforce mathematics covered. Students and teachers of undergraduate mathematics courses will find this volume a first-rate introduction to algebraic number theory.

User’s Reviews

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

⭐This is a good book on number theory. It starts quickly with the fundamental theorems and proofs, far more directly and simply than I’ve seen in other number theory books. It’s an older book and gives the feel of high quality mathematics writing, clear explanations, and beautiful proofs, and lots of nice problems.

⭐For the experienced mathematical person this book is very relevant and nice to start with when digging in algebraic number theory. Basic knowledge of algebraic number theory is also in the book coupled to a lot of exercises.

⭐Started reading it but it sure is technical, need to really think these through

⭐This review refers to the 1965 Hardcover version of the book.It’s quite apparent that the 40 years that have passed since this book was printed have very much dated it’s content. The definitions of many key concepts (such as an ideal) contain the right ideas, but are not formulated in the modern viewpoint. These, however are only minor setbacks. The main flaw of this book is its subject matter. There are 11 chapters, and it was not until the eighth that the ideas start getting deeper. Even these last 4 chapters do not delve very far into the heart of things.The text is written with the reader in mind (almost excessively so). Useful equations are clearly labeled and the steps in the proof are clearly outlined, though sometimes to an unnecessary degree.I would recommend this book for a mathematics hobbyist, or perhaps an undergraduate number theory course. For anyone with a stronger background, they wil not glean much.

⭐Proceeding from the Fundamental Theorem of Arithmetic, into Fermat’s Theory for Gaussian Primes, this book provides a very strong introduction for the advanced undergraduate or beginning graduate student to algebraic number theory. The book also covers polynomials and symmetric functions, algebraic numbers, integral bases, ideals, congruences and norms, and the UFT.

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