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Though Mariano's comment above is no doubt true and the most complete answer you'll get, there are a couple of texts that stand apart in my mind from the slew of textbooks with the generic title "Algebraic Number Theory" that might tempt you. The first leaves off a lot of algebraic number theory, but what it does, it does incredibly clearly and it's cheap! It's "Number Theory I: Fermat's Dream", a translation of a Japanese text by Kazuya Kato.
For something a little more encyclopedic after you're done with those if it's possible to be "done" with Cox's book , my personal favorite more comprehensive reference is Neukirch's Algebraic Number Theory. Marcus's Number Fields is a good intro book, but its not in Latex, so it looks ugly. Also doesn't do any local p-adic theory, so you should pair it with Gouvea's excellent intro p-adic book and you have great first course is algebraic number theory.
Many people have recommended Neukirch's book. I think a good complement to it is Janusz's Algebraic Number Fields. They cover roughly the same material. Neukirch's presentation is probably the slickest possible; Janusz's is the most hands on. I love them both now, but I found Janusz understandable at a point when Neukirch was still completely impenetrable.
I would recommend you take a look at William Stein's free online algebraic number theory textbook. It is especially useful if you want to learn how to compute with number fields, but it is still extremely readable even if you skip the details of the computational examples. I could be wrong, but I think Borevich and Shafarevich cover material related to Pell's equation. If not, then it is still an excellent book on algebraic number theory as is Serre's "A Course in Arithmetic". However Serre does not discuss Pell's equation. In particular, in view of the focus of your studies, I suggest the following additional book; where additional is meant that I would not suggest it as the only book see below for explanation.
It contains material related to Diophantine equations and the tools used to study them, in particular, but not only, those from Algebraic Number Theory. Yet, this is not really an introduction to Algebraic Number Theory; while the book contains a chapter Basic Algebraic Number Theory, covering the 'standard results', it does not contain all proofs and the author explictly refers to other books including several of those already mentioned.
However, I could imagine that a rich exposition of how the theory you are learning can be applied to various Diophantine problems could be valuable. If you want to have a pretty solid foundation of this subject, then you are suggested to read the book Lectures on Algebraic Number Theory by Hecke which is extremely excellent in the discussion of topics even important nowadays, or the report of number theory by Hilbert whose foundation is indeed solid. In addition, Gauss's book, being a little old and hard, is a good reference on quadratic forms and it itself offers two different kinds of proofs of the quadratic reciprocity law which are all excellent to me.
The last but not the least, I would like to confirm once more the book by Jurgen Neukirch which notes the connection between ideals and lattices, i. Map of Number Theory. See Solving the Pell Equation reviewed here. You probably know Lenstra Jr. It has been recently reprinted by the LMS.
I find it to be one of the most clear math books on an advanced topic, ever. The book Number theory II by Koch translated by Parshin and Shafarevich is very good, and contains some hard-to-find material. For example, they give a presentation of the absolute Galois group of a local field. By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service , privacy policy and cookie policy , and that your continued use of the website is subject to these policies.
Some sections are terse, and an instructor may want to supplement the theoretical exercises with some more computational ones. The Euclidean Algorithm is presented after sections on solving linear congruences modulo n, and the Chinese Remainder Theorem; applications of the Euclidean Algorithm to these topics are presented later.
An instructor may want students to become comfortable with these topics initially through computations, using the Euclidean Algorithm. We should also point out that mathematical induction is a prerequisite for this text, and some of the material is presented using pseudocode, which is different than many texts on these topics. Since the book is quite long, an instructor for a one-semester course would need to choose specific topics from the text to cover.
There are a few sections indicated that are not required for future material. However, even among the remaining sections, an instructor would need to carefully choose sections that include all necessary prerequisite material.
Depending on a course's focus, this could be done fairly easily. Each chapter is written in a logical manner, referencing previous material as needed.
The book jumps from chapters on purely algebraic topics to those focused on applications. For this reason, an instructor may want to choose certain sections in a chapter to cover as prerequisites for an application, instead of covering the material linearly.
As promised by the title, the book gives a very nice overview of a side range of topics in number theory and algebra primarily the former, but with quite a bit of attention to the latter as well , with special emphasis to the areas in which As promised by the title, the book gives a very nice overview of a side range of topics in number theory and algebra primarily the former, but with quite a bit of attention to the latter as well , with special emphasis to the areas in which computational techniques have proved useful. There is a very good index and glossary and a good review of notation and basic facts in the first chapter.
The format of the book makes it especially easy to update as advances in the subjects occur, particularly computational advances. References are given to websites as well as books.
The prose is very lucid and easy to follow. Many examples are given and difficult ideas are introduced gradually. The many relationships between number theory and algebra are explored in detail, each subject yielding important insights into and applications of the other. No jargon is used and terminology is carefully explained.
The book has a very consistent framework and a nice flow from one chapter to the next. As mentioned above, relationships between the two subjects of the title are emphasized. The book is nicely broken up into manageable sections that would fit well into a lecture course. Interdependences among chapters are clearly indicated. The topics are presented clearly and logically with relationships among them clearly pointed out and discussed in detail.
All pages display very well on my screen, with no legibility or distortion issues that I could see. This is not relevant for a mathematics text, but I saw nothing that would be offensive to a reader of any ethnic background. The text is so comprehensive that it feels overwhelming. The author wanted to include all of the mathematics required beyond a standard calculus sequence.
However, the mathematical maturity required to read and learn from this text is quite However, the mathematical maturity required to read and learn from this text is quite high. The first two chapters cover much of a standard undergraduate course in number theory, built up from scratch. However, it almost completely lacks numerical examples and computational practice for the students, which would give those new to the material time and experience in which to digest, assimilate, and understand the material.
I would think that a book targeted at this level of mathematical sophistication would assume students are comfortable with for example the most basic notions of group theory or the idea of equivalence classes. I can't imagine an appropriate audience for this text: I found no mathematical errors. The mathematical presentation is rigorous, clear, and well-explained.
It can be terse at times, skipping steps and making conceptual leaps that will be challenging for all but the very best students. The book covers both standard background that will always be relevant for these topics: The computational chapters use pseudocode, so they will not be quickly outdated when new languages become fashionable.
Most of the algorithms studied are quite "classical" as much as that makes sense for computer science , with modern ideas and developments usually relegated to "Notes" at the end of the computational chapters. This will, of course, become outdated with new research in computer science. But any faculty member who keeps up with the relevant research will be able to mention new developments to students, and it will not interrupt the flow of the ideas at all.
The book is exceedingly well written, though it is at a very high level. It is not "friendly" or "chatty" as you will find with many number theory books targeted to undergraduates. For many students this will detract from clarity because they do not yet have the mathematical sophistication to work at this level. The book does an excellent job of consistency of notation. For example, it starts with a development of number theory concepts, and develops notation for residue classes in the integers modulo n.
Later in the chapters on groups and rings, this same notation is used in more general situations.
Whenever there is the potential for confusion for example, in using "a mod b" as a binary operation as is common in computer science versus using "a is congruent to x mod b" as is more standard in mathematics the author is careful to point out the dual meanings and to warn the reader that there is some overloading of terminology. It is unavoidable that this will happen in any book that treats both subjects seriously, and the author is careful with notation and keeps potential confusion to a minimum.
The book has 21 chapters, each with several sections. Most, but not all, sections end with a set of exercises. Essential exercises are underlined a very nice feature!
What would be helpful would be some suggested paths through the text for various purposes. I don't think it would be appropriate in any class to start at Chapter 1 and and work through all or even most of the content. I imagine that most classes would skip the background material and head straight for the computational chapters, with the background there "as needed" for the students.