I have just finished a master's degree in mathematics and want to learn everything possible about algebraic number fields and especially applications to the generalized Pell equation (my thesis topic), $x^2Dy^2=k$, where $D$ is square free and $k \in \mathbb{Z}$. I have a solid foundation in modern algebra and elementary number theory as well as analysis. Does anyone have any suggestions? I am currently reading Harvey Cohn's 'Advanced number theory' with slow but marked progress. Thanks.

51$\begingroup$ To learn 'everything possible' about algebraic number fields you'll have to reserve, say, your next 500 years... $\endgroup$– Mariano SuárezÁlvarezJan 28 '10 at 21:24

20$\begingroup$ Ersnt Kunz starts the forword to his Introduction to Commutative Algebra and Algebraic Geometry with: «It has been estimated that, at the present state of our knowledge, one could give a 200 semester course on commutative algebra and algebraic geometry without ever repeating himself.» His subject is not unique in that respect! $\endgroup$– Mariano SuárezÁlvarezJan 28 '10 at 21:53

9$\begingroup$ Pierre Samuel's "Algebraic Theory of Numbers" gives a very elegant introduction to algebraic number theory. It doesn't cover as much material as many of the books mentioned here, but has the advantages of being only 100 pages or so and being published by dover (so that it costs only a few dollars). Reading this would certainly prepare you well for some of the more advanced books that require more of a commitment to go through. $\endgroup$– user1073Feb 1 '11 at 20:59

5$\begingroup$ Cohn's book is well worth reading carefully, and Ireland and Rosen is an excellent text too. $\endgroup$– EmertonFeb 2 '11 at 15:09

3$\begingroup$ @RaphaelAlbrecht, you will agree that there has been exactly one Grothendieck in a long, long, looooong time. $\endgroup$– Mariano SuárezÁlvarezJan 28 '14 at 20:12
I know of very few more endearing books on the subject than Ireland and Rosen's A Classical Introduction to Modern Number Theory.

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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. The second is Cox's "Primes of the form $x^2+ny^2$, which in terms of getting to some of the most amazing and deepest parts of algebraic number theory with as few prerequisites as possible, has got to be the best choice. 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.

7$\begingroup$ Agreed. Neukirch is an amazing book, not least because of how seriously it takes the analogy between number fields and function fields. $\endgroup$ Jan 28 '10 at 23:33

1$\begingroup$ Were the other books in Kato's series every translated? $\endgroup$ Jul 7 '10 at 4:54

1$\begingroup$ Number Theory 2 was just released by the AMS! $\endgroup$ Nov 9 '11 at 13:38
Marcus's Number Fields is a good intro book, but it's not in LaTeX, so it looks ugly. Also doesn't do any local (padic) theory, so you should pair it with Gouvêa's excellent intro padic book and you have a great first course in algebraic number theory.
Update: a typeset version of Marcus was published in 2018 and a third edition of Gouvêa was published in 2020.

8$\begingroup$ Yes, someone should typeset Marcus's book again in LaTeX. $\endgroup$– lhfJan 29 '10 at 1:11

4$\begingroup$ Marcus wrote the book much before Wiles proved the FLT; so the introductory chapter on solving FLT for regular primes etc is fascinating. Also, the book has lots of concrete problems and exercises. $\endgroup$ Jul 7 '10 at 5:31

4$\begingroup$ Marcus' book is my first choice. Very handson, many exercises, and clear explanations. More than makes up for the fact that it's typewritten. Once you have understood something, you can then go and read about it again in Neukirch (which covers a lot more stuff), it will give you a different perspective. $\endgroup$ Feb 1 '11 at 18:39

3$\begingroup$ +1 For Marcus, really one of the best books on the subject. And yes, you are not the only people thinking that it deserves to be retyped! $\endgroup$ Feb 1 '11 at 19:36

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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.
Neither of these is particularly strong on the Pell equation, though.
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.

$\begingroup$ Seems like a good option for cryptographers. $\endgroup$ Dec 15 '20 at 19:43

1$\begingroup$ @AryaPourtabatabaie The first rule of cryptography club is 9088022396507725372745554020228708407452. $\endgroup$ Mar 5 at 10:10
It might be at too basic/introductory a level for your purposes, but as an undergraduate I really liked Stewart and Tall's Algebraic Number Theory (2nd edition?)

$\begingroup$ An undergraduate book worth mentioning is Algebraic Number Theory by Jarvis (see here for a review). I've put this as a comment, as the text might be too gentle for the OP and many other readers on MO, but could be a great choice for an advanced undergraduate. $\endgroup$– J WMar 5 at 15:16


2$\begingroup$ @JW I'm aware of it; I think I looked briefly at it; I decided I preferred the 2nd edition :D (Both in terms of content but also in terms of the physical published book; I miss Chapman and Hall / CRC before they got acquired by Taylor+Francis $\endgroup$ Mar 5 at 17:59
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).
There is a fairly recent book (in two volumes) by Henri Cohen entitled "Number Theory" (Graduate Texts in Mathematics, Volumes 239 and 240, Springer). [To avoid any risk of confusion: these are not the two GTMbooks by the same author on computational number theory.]
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.
Final note: the book is in two volumes, the second one is mainlyon analytic tools, linear forms in logarithms and modular forms applied to Diophantine equations; for the present context (or at least initially), the first volume is the relevant one.
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.
(I also found Cohn difficult to read =)
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.e. algebraic numbers and geometry.
Look at: Map of Number Theory
The book recommended there (Manin/Panchishkin's "Introduction to Modern Number Theory") does seem amazingly comprehensive, and very readable.
I am amazed not to see the book "Introductory Algebraic Number Theory," by Alaca & Williams, listed here. I find it to be one of the most clear math books on an advanced topic, ever.
Serge Lang's Algebraic Number Theory has a lot of general theoretical material.
If you want to learn class field theory (which you should at some point, after you have read an introductory book on algebraic number theory), then "Algebraic Number Theory" edited by Cassels and Fröhlich is a classic that doesn't get old. It has been recently reprinted by the LMS.
See Solving the Pell Equation (reviewed here). You probably know Lenstra Jr.'s article in the AMS Notices. There's also Primes of the form $x^2 + ny^2$ by Cox.
The book Number theory II by Koch (translated by Parshin and Shafarevich) is very good, and contains some hardtofind material. For example, they give a presentation of the absolute Galois group of a local field.
Also (I can't imagine someone else hasn't suggested this) the conference proceedings Algebraic number theory, edited by Cassels and Fröhlich, is a pretty standard text with lots of useful stuff (including Tate's thesis!)

$\begingroup$ Cassels and Fröhlich has been suggested by Alex B. $\endgroup$ Nov 1 '13 at 16:40