most recent 30 from http://mathoverflow.net 2013-05-22T07:49:47Z http://mathoverflow.net/feeds/question/8097 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8097/number-theory-textbook-with-an-algebraic-perspective Harry Gindi 2009-12-07T10:11:41Z 2010-01-28T21:44:58Z

<p>Most of the number theory textbooks I've dealt with take a very classical approach to the subject. I'm looking for a textbook that's something like a first course in number theory for people who have a decent command of modern algebra (at the level of something like Lang's Algebra). Does such a book exist, and if it does, what is it called?</p> <p>Edit: As I posted in a comment below:</p> <p>In the introduction to Ireland and Rosen, they note something that was bugging me for a while, "Nevertheless it is remarkable how a modicum of group and ring theory introduces unexpected order into the subject." </p> <p>This is precisely the perspective I was looking for, so if anyone passes by this topic looking for a book that approaches number theory in this way, I feel like this quote should point him (her?) in the right direction. </p>

http://mathoverflow.net/questions/8097/number-theory-textbook-with-an-algebraic-perspective/8099#8099 lhf 2009-12-07T10:38:18Z 2009-12-07T10:38:18Z

<p>There are probably many such books, for instance "<a href="http://store.doverpublications.com/0486689069.html" rel="nofollow">Fundamentals of Number Theory</a>" by LeVeque, "<a href="http://store.doverpublications.com/0486458075.html" rel="nofollow">Elementary Number Theory</a>" by Bolker and "<a href="http://www.springer.com/math/numbers/book/978-0-387-97329-6" rel="nofollow">A Classical Introduction to Modern Number Theory</a>" by Ireland and Rosen.</p>

http://mathoverflow.net/questions/8097/number-theory-textbook-with-an-algebraic-perspective/8100#8100 Jose Brox 2009-12-07T10:49:44Z 2009-12-07T10:49:44Z

<p>Well, it depends on the actual subject you want to approach and the "decent command of modern algebra" already assumed; without knowing more, I would recommend:</p> <ul> <li><p><em>Number fields</em> (1995) by <strong>Marcus</strong>. Universitext. Just as the title says, a (great!) introduction to number fields. </p></li> <li><p><em>Algebraic Number Theory</em> (1986) by <strong>Cassels</strong> and <strong>Frölich</strong>. Academic Press. It explains the basics (class field theory, zeta functions) to understand the Langlands Program.</p></li> <li><p><em>A course on arithmetic</em> (1996) by <strong>Serre</strong>. Graduate Texts in Mathematics. P-adic fields, quadratic forms, zeta functions and modular forms.</p></li> </ul>

http://mathoverflow.net/questions/8097/number-theory-textbook-with-an-algebraic-perspective/8101#8101 Jose Brox 2009-12-07T12:33:00Z 2009-12-07T12:33:00Z

<p>If I understand it well now, what you want are books about <strong>basic</strong> number theory with a good algebraic foundation. I can recommend the following:</p> <ul> <li><p><em>Elementary methods in number theory</em> (2000), by <strong>Nathanson</strong>. Graduate Texts in Mathematics. It starts low, but it reaches quite high.</p></li> <li><p><em>Elementary Number Theory with Applications</em> (2007) by <strong>Koshy</strong>. Elsevier. Truly basic. Not very very algebraic, but a really nice textbook.</p></li> <li><p><em>Algebra and number theory</em> by Andrew <strong>Baker</strong>. Online notes. Fairly basic.</p></li> <li><p><em>A computational introduction to number theory and algebra</em> (2005) by <strong>Shoup</strong>. Cambridge University Press / Online Free Version. Despite the title, I think it satisfies the conditions you are looking for.</p></li> </ul>

http://mathoverflow.net/questions/8097/number-theory-textbook-with-an-algebraic-perspective/13286#13286 Felipe Voloch 2010-01-28T21:44:58Z 2010-01-28T21:44:58Z

<p>Borevich-Shafarevich</p>