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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?

Edit: As I posted in a comment below:

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."

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.

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  • $\begingroup$ fpqc--If you'd like to come by my office and take a look at some of these books, I have many of them on my shelf, feel free to send me an e-mail and stop by. $\endgroup$
    – Ben Weiss
    Commented Dec 7, 2009 at 20:42

4 Answers 4

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There are probably many such books, for instance "Fundamentals of Number Theory" by LeVeque, "Elementary Number Theory" by Bolker and "A Classical Introduction to Modern Number Theory" by Ireland and Rosen.

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  • $\begingroup$ Do you have a recommendation of one in particular? $\endgroup$
    – Harry Gindi
    Commented Dec 7, 2009 at 10:39
  • $\begingroup$ I like all three; LeVeque is a good and cheap, though I'm not sure what level you aim at. $\endgroup$
    – lhf
    Commented Dec 7, 2009 at 10:43
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    $\begingroup$ I'm looking through Ireland and Rosen right now, and it seems to be what I was looking for. $\endgroup$
    – Harry Gindi
    Commented Dec 7, 2009 at 10:51
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    $\begingroup$ I like Ireland-Rosen book very much too. $\endgroup$
    – mathreader
    Commented Dec 7, 2009 at 11:21
  • $\begingroup$ Seconded. I always wish to have read Ireland-Rosen much earlier. $\endgroup$
    – user709
    Commented Dec 7, 2009 at 13:40
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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:

  • Number fields (1995) by Marcus. Universitext. Just as the title says, a (great!) introduction to number fields.

  • Algebraic Number Theory (1986) by Cassels and Frölich. Academic Press. It explains the basics (class field theory, zeta functions) to understand the Langlands Program.

  • A course on arithmetic (1996) by Serre. Graduate Texts in Mathematics. P-adic fields, quadratic forms, zeta functions and modular forms.

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  • $\begingroup$ I was looking not for a book on algebraic number theory, but an algebraic book on number theory, if you understand the distinction I'm trying to make. $\endgroup$
    – Harry Gindi
    Commented Dec 7, 2009 at 10:53
  • $\begingroup$ Not really. Can you elaborate, please? Maybe I can suggest some other kind of books! $\endgroup$
    – Jose Brox
    Commented Dec 7, 2009 at 11:33
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    $\begingroup$ 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." This is precisely the perspective I was looking for. $\endgroup$
    – Harry Gindi
    Commented Dec 7, 2009 at 11:50
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    $\begingroup$ I wish Marcus's "Number fields" had a new edition, typeset, not typewritten. Otherwise, it's a great book with lots of hands-on material. $\endgroup$
    – lhf
    Commented Dec 7, 2009 at 15:19
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    $\begingroup$ @lhf, the good news is that, after many years, there is a newly-typeset edition of Marcus: link.springer.com/book/10.1007/978-3-319-90233-3 (Springer 2018) $\endgroup$
    – J W
    Commented Jan 2, 2022 at 15:28
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Borevich-Shafarevich

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If I understand it well now, what you want are books about basic number theory with a good algebraic foundation. I can recommend the following:

  • Elementary methods in number theory (2000), by Nathanson. Graduate Texts in Mathematics. It starts low, but it reaches quite high.

  • Elementary Number Theory with Applications (2007) by Koshy. Elsevier. Truly basic. Not very very algebraic, but a really nice textbook.

  • Algebra and number theory by Andrew Baker. Online notes. Fairly basic.

  • A computational introduction to number theory and algebra (2005) by Shoup. Cambridge University Press / Online Free Version. Despite the title, I think it satisfies the conditions you are looking for.

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  • $\begingroup$ Numbers one and three look interesting, while numbers two and four look awful. $\endgroup$
    – Harry Gindi
    Commented Dec 7, 2009 at 12:36

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