Download PDF Number Theory: An Introduction via the Distribution of Primes, by Benjamin Fine, Gerhard Rosenberger
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Number Theory: An Introduction via the Distribution of Primes, by Benjamin Fine, Gerhard Rosenberger
Download PDF Number Theory: An Introduction via the Distribution of Primes, by Benjamin Fine, Gerhard Rosenberger
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This book provides an introduction and overview of number theory based on the distribution and properties of primes. This unique approach provides both a firm background in the standard material as well as an overview of the whole discipline. All the essential topics are covered: fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. Analytic number theory and algebraic number theory both receive a solid introductory treatment. The book’s user-friendly style, historical context, and wide range of exercises make it ideal for self study and classroom use.
- Sales Rank: #2370117 in Books
- Published on: 2010-06-02
- Original language: English
- Number of items: 1
- Dimensions: 9.25" h x .82" w x 6.10" l, 1.12 pounds
- Binding: Paperback
- 342 pages
Review
This book attempts the rather ambitious task of covering much of elementary, analytic and algebraic number theory in a bit over 300 pages. –MathSciNet
From the Back Cover
This book provides an introduction and overview of number theory based on the distribution and properties of primes. This unique approach provides both a firm background in the standard material as well as an overview of the whole discipline. All the essential topics are covered: fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes.
Key Topics and Features:
* Solid introduction to analytic number theory, including full proofs of Dirichlet’s Theorem and the Prime Number Theorem
* Solid treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals
* First treatment in book form of the AKS algorithm that shows that primality testing is of polynomial time
* Many interesting side topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers
The book’s user-friendly style, historical context, and wide range of exercises from simple to quite difficult (with solutions and hints provided for select ones) make it ideal for self study as well as classroom use. Intended for upper level undergraduates and beginning graduate students, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced in the book.
Most helpful customer reviews
5 of 5 people found the following review helpful.
Fantastic Second Book on Number Theory
By M.J. Headlee
This book is an excellent second book to Number Theory. If you've already had number theory, you will have seen the topics they are presenting in bits. What this book gives more than anything is CONTEXT. That context is very valuable and can help direct you to future readings.
Most other number theory books are rather ad-hoc. You do some congruences, then you see some quadratic reciprocity, then maybe some neat facts about divisors and perfect numbers. Not so here. While Fine and Rosenburger do tackle those topics, everything is framed in the context of prime numbers. In particular, they take pains to introduce specifically (i) analytic number theory (Including Dirichlet's proof et the prime number theorem), (ii) algebraic number theory, and (iii) additive number theory (Waring problem). They also have a section on cryptography and a proof of the AKS algorithm. There is an entire chapter devoted to nearly 15+ proofs that there are infinitely many primes, using basic techniques as well as more advanced methods from topology and analysis. That alone makes the book a gem.
This book seems better suited as a second book in number theory because of the perspective they take. They look down on number theory from Modern Algebra to give their sweeping panorama. While they do have presentations on things like Ring Theory in the book, if you are seeing that for the first time with this book you will be busy digesting Ring Theory instead of appreciating that perspective. You can also see the context of all the other "random things" that prior books have done.
Overall, a fantastic reference I am glad to have on my shelf. But I wouldn't want to start from here.
5 of 5 people found the following review helpful.
Very good, but ...
By Barney Sperlin
Though I'm still near the beginning of the book I've been put off by the large number of typos and incorrect statements. It's not hard to work around them, but it is irritating and makes you wish for a good editor to have taken this prior to our reading it. For example on p. 8 there is a range error after Lemma 2.1.1 and on p. 11, Theorem 2.2.1 part (5) contradicts part (4). There is also an error in the commentary later in that page.
The point made by the earlier poster concerning the benefits of context is excellent and certainly saves the book.
0 of 0 people found the following review helpful.
Five Stars
By Daniel Mario Valdez
all ok
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