most recent 30 from http://mathoverflow.net 2013-05-22T20:37:08Z http://mathoverflow.net/feeds/question/13282 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books Jason Smith 2010-01-28T21:22:35Z 2013-03-10T14:14:23Z

<p>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), $x^2-Dy^2=k$, 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.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/13284#13284 Yemon Choi 2010-01-28T21:37:59Z 2010-01-28T21:37:59Z

<p>It might be at too basic/introductory a level for your purposes, but as an undergraduate I really liked Stewart and Tall's <em>Algebraic Number Theory</em> (2nd edition?)</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/13285#13285 Mariano Suárez-Alvarez 2010-01-28T21:42:13Z 2010-01-28T21:42:13Z

<p>I know of very few more endearing books on the subject than Ireland and Rosen's <em>A Classical Introduction to Modern Number Theory</em>.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/13289#13289 Cam McLeman 2010-01-28T21:47:57Z 2010-01-28T21:47:57Z

<p>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.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/13294#13294 Randy Brown 2010-01-28T22:44:14Z 2010-01-28T22:44:14Z

<p>Look at: <a href="http://mathoverflow.net/questions/13106/map-of-number-theory" rel="nofollow">http://mathoverflow.net/questions/13106/map-of-number-theory</a></p> <p>The book recommended there (Manin/Panchishkin's "Introduction to Modern Number Theory") does seem amazingly comprehensive, and very readable.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/13298#13298 Akhil Mathew 2010-01-28T23:18:32Z 2010-01-28T23:18:32Z

<p>Serge Lang's <em>Algebraic Number Theory</em> has a lot of general theoretical material.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/13299#13299 Alison Miller 2010-01-28T23:28:09Z 2010-01-28T23:28:09Z

<p>I would recommend you take a look at William Stein's <a href="http://sage.math.washington.edu/books/ant/ant.pdf" rel="nofollow">free online algebraic number theory textbook</a>. 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.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/13304#13304 Rado 2010-01-29T00:12:41Z 2010-01-29T00:12:41Z

<p>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.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/13315#13315 lhf 2010-01-29T01:05:47Z 2010-01-29T01:05:47Z

<p>See <a href="http://www.springer.com/math/numbers/book/978-0-387-84922-5" rel="nofollow">Solving the Pell Equation</a> (reviewed <a href="http://www.maa.org/maa%20reviews/7307.html" rel="nofollow">here</a>). You probably know Lenstra Jr.'s <a href="http://www.ams.org/notices/200202/fea-lenstra.pdf" rel="nofollow">article</a> in the AMS Notices. There's also <a href="http://www.cs.amherst.edu/~dac/primes.html" rel="nofollow">Primes of the form $x^2 + ny^2$</a> by Cox.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/30856#30856 kh 2010-07-07T04:43:42Z 2010-07-07T04:43:42Z

<p>I could be wrong, but I think <a href="http://books.google.com/books?id=njgVUjjO-EAC&amp;printsec=frontcover&amp;dq=borevich+shafarevich&amp;source=bl&amp;ots=ciicW0mupa&amp;sig=UlX3d4H-z4OTFPXZrmCldBkDXGI&amp;hl=en&amp;ei=PQU0TPbFM8H78Aa-jaG6Aw&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=3&amp;ved=0CB8Q6AEwAg#v=onepage&amp;q&amp;f=false" rel="nofollow">Borevich and Shafarevich</a> 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.</p> <p>(I also found Cohn difficult to read =)</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/53994#53994 awllower 2011-02-01T15:49:42Z 2011-02-02T13:33:16Z

<p>If you want to have a <strong>pretty solid</strong> foundation of this subject, then you are suggested to read the book <strong><em>Lectures on Algebraic Number Theory</em></strong> 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.<br> 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.<br> The last but not the least, I would like to confirm once more the book by <strong><em>Jurgen Neukirch</em></strong> which notes the connection between ideals and lattices, i.e. algebraic numbers and geometry.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/54014#54014 quid 2011-02-01T19:21:22Z 2011-02-01T19:21:22Z

<p>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).</p> <p>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 GTM-books by the same author on computational number theory.]</p> <p>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). </p> <p>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.</p> <p>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. </p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/54016#54016 David Speyer 2011-02-01T19:43:22Z 2011-02-01T19:43:22Z

<p>Many people have recommended Neukirch's book. I think a good complement to it is Janusz's <i>Algebraic Number Fields</i>. 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.</p> <p>Neither of these is particularly strong on the Pell equation, though.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/54102#54102 Alex Bartel 2011-02-02T15:20:24Z 2011-02-02T15:20:24Z

<p>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 <a href="http://mathfit.ac.uk/CasselsFrohlich.pdf" rel="nofollow">recently reprinted</a> by the LMS.</p>

http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books/124126#124126 J K 2013-03-10T04:34:27Z 2013-03-10T04:34:27Z

<p>I am amazed not to see the book "Introductory Algebraic Number Theory," by Alaca &amp; Williams, listed here. I find it to be one of the most clear math books on an advanced topic, ever.</p>