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A Computational Introduction to Number Theory and Algebra (shoup.net)
163 points by luu on Aug 26, 2015 | hide | past | favorite | 17 comments



Algebraic Number Theory: A Computational Approach http://wstein.org/books/ant/ant.pdf

A Course in Computational Algebraic Number Theory http://bit.ly/1heah8l


Author of "Algebraic Number Theory: A Computational Approach" here, in case anybody has any questions. Here's the history of that book. I first taught an undergraduate class at Harvard in maybe 2002 and went over the first 20 pages of Swinnerton-Dyer's brief course on algebraic number theory book -- expanding it into course-length notes. I taught the course next to grad students at UC San Diego, and added more content inspired by the excellent "Algebraic Number Theory" by Cassels-Frohlich. Then I taught it again twice at Univ of Washington, adding more modern computational content, and resulting in a rough draft of this book. Finally, Travis Scholl (a UW grad student) and I spent the last year polishing it and making it look a bit nicer. The book is under contract to be published by the American Mathematical Society soon.


Thanks for putting this out there for free! I think its an amazing thing to do, especially for the more academic books. Anecdotally, my friend, who is now a mathematics grad student in University of Western Ontario, learned mathematics entirely from ebooks and the low price editions that you find in India.


Thank you for such a great work and especially for making it available for free! For anyone interested, I can also recommend trying out Sage! There is also this free book (http://abstract.pugetsound.edu/sage-aata.html) on Abstract Algebra which includes Sage exercises. The most prominent application of number theory in computer science is probably cryptography. I can highly recommend "An Introduction to Mathematical Cryptography" by Hoffstein, Pipher and Silverman in that area.


Not saying the book isn't good, but I have a general observation to make. Such books would be better if they provided a sequence of carefully difficulty-graded exercises that would build on towards practical mastery. Instead, there is a flood of theorems with a spattering of exercises. With two exercises one can't feel confident about learning a theorem or other deep complex math concept.

It's one thing to read a theorem, another to be confident to apply it. When I was learning math in university, it was the same. Theorems, axioms and definitions by the truckload, but exercises - nada. In reality it all comes down to applying math.


What background of mathematics does this book assume?


From the preface:

Prerequisites. The mathematical prerequisites are minimal: no particular mathematical concepts beyond what is taught in a typical undergraduate calculus sequence are assumed.

The computer science prerequisites are also quite minimal: it is assumed that the reader is proficient in programming, and has had some exposure to the analysis of algorithms, essentially at the level of an undergraduate course on algorithms and data structures.

Even though it is mathematically quite self contained, the text does presuppose that the reader is comfortable with mathematical formalism and also has some experience in reading and writing mathematical proofs. Readers may have gained such experience in computer science courses such as algorithms, automata or complexity theory, or some type of “discrete mathematics for computer science students” course. They also may have gained such experience in undergraduate mathematics courses, such as abstract or linear algebra. The material in these mathematics courses may overlap with some of the material presented here; however, even if the reader already has had some exposure to this material, it nevertheless may be convenient to have all of the relevant topics easily accessible in one place; moreover, the emphasis and perspective here will no doubt be different from that in a traditional mathematical presentation of these subjects


What would be a good textbook for Math 101, specifically to learn some advanced mathematical formalism without actually diving in applied science behind it?


Keeping with the free theme, Book Of Proof by Richard Hammack is a nice introduction to proofs and formalism. It's available free from the author as a PDF[1], and also as a physical book on Amazon[2].

An alternative if you're willing to spend a little is How to Prove It by Daniel J. Velleman, also available from Amazon[3] and probably many other retailers. Both books cover roughly the same topics.

[1]: http://www.people.vcu.edu/~rhammack/BookOfProof/

[2]: http://www.amazon.com/Book-Proof-Richard-Hammack/dp/09894721...

[3]: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/05...


This stuff is usually rolled into courses called "abstract algebra". If a textbook is called "abstract algebra", it's usually designed for a first year undergraduate. If it's just called "algebra", it's usually aimed at a more mature audience. Herstein and Hungerford are the texts I learned from. This book seems more recent and popular:

http://www.amazon.com/Book-Proof-Richard-Hammack/dp/09894721...

The other big stream of basic undergraduate mathematics is analysis. For that I recommend Spivak's Calculus.


Coursera's Calculus One and Calculus Two courses.

I've found other branches of math to be much easier to understand if you have basic knowledge about calculus.


I really like Hubbard & Hubbard.


From the third review hyperlink (first two are dead): https://amca01.wordpress.com/2009/10/03/book-review-a-comput...

"Although the text requires not much specific mathematical background, I would hesitate to use it except in an advanced class, or for students whose mathematical ability was already high. The material moves swiftly – while never compromising rigour – and the multiple strands assume considerable ability on the part of the reader."

Math textbooks must be reviewed by an expert; yet it's impossible for an expert to see them as a beginner would. If they can see it's difficult for a beginner, it definitely is...


I read a bit the chapter about the distribution of prime numbers. I could follow because I have an Msc in maths (not in number theory) but the progression is fast, and while I don't think one needs an advanced background to read it, one needs to be used to formal reasoning, and I feel like the author is moving fast, so I guess for someone with a low background in maths it would be a hard (but not impossible) read.



How relevant is this book to artificial intelligence


Depends what you mean by AI. Back in the day, a computer algerbra system was considered AI, in which case, this book is very relevant. But I can't think of another interpretation of AI that would make this book relevant.




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