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I have read some elementary number theory from David Burton's text and I know groups and rings from Herstein's book Topics in Algebra and some field theory from different sources online. I am currently learning commutative algebra from Atiyah and Macdonald's engaging book on commutative algebra.

I would really like to learn some algebraic number theory and I was wondering if someone could provide me a sequence of books/ steps so that I can understand enough number theory to be a strong graduate student about to begin research. I apologize sincerely if my question is off-topic here or partly vague.

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  • $\begingroup$ mathoverflow.net/questions/155309/… $\endgroup$ – Ben McKay May 30 '16 at 20:23
  • $\begingroup$ You might want to make this community wiki, since you're likely to get many different answers with no one answer being the correct one. $\endgroup$ – David White May 30 '16 at 20:24
  • $\begingroup$ @DavidWhite, I am not able to make it community wiki. There is no option available. $\endgroup$ – user96343 May 30 '16 at 20:27
  • $\begingroup$ Okay, I flagged it so the moderators can do it. In the meanwhile, have you seen: math.stackexchange.com/questions/176552 and math.stackexchange.com/questions/66086/… $\endgroup$ – David White May 30 '16 at 20:41
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    $\begingroup$ I'm not a number theorist at all, but my friends who are all started with Class Field Theory. I'm sure this depends on the type of research done wherever you go to grad school, so maybe the best advise is to get accepted somewhere, then ask the professors there. I remember the book my friends read was old, yellow, and thin. Perhaps it's the one by Neukirch. If you don't want to wait till being accepted somewhere, why not follow a syllabus from online? For instance, here is one from the Budapest program: bsmath.hu/12fall/NUT3syllabus.html $\endgroup$ – David White May 31 '16 at 14:24
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Two good books for an introduction to global algebraic number theory (i.e., number fields) are:

Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger (Dover Books on Mathematics) May 19, 2008 by Pierre Samuel (less than $8 in paperback)

A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) by Kenneth Ireland & Michael Rosen, Springer

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    $\begingroup$ I used Samuel as a text when I taught AlgNoTh way back when, and I liked it a lot. Still dip into it, too. $\endgroup$ – Lubin Jun 4 '16 at 15:42
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    $\begingroup$ @Lubin Indeed you did. And Susan and I were in that class when you taught it in spring 1977!! (Except I think that it was actually the second semester graduate algebra course, since there was no separate graduate number theory course back then.) $\endgroup$ – Joe Silverman Jun 4 '16 at 17:08
  • $\begingroup$ Oh my, was it that long ago? $\endgroup$ – Lubin Jun 4 '16 at 17:10
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Number theory is broad, and I don't think it's something one learns sequentially. (Can I say there's no royal road?) There are lots fundamental topics, such as:

  • number fields
  • p-adic numbers
  • quadratic forms
  • elliptic curves
  • modular forms
  • class field theory
  • zeta functions
  • L-functions

Most of these topics (all but CFT) you could start learning first, and there are lots of ways to learn them.

Here are some some suggestions for relatively gentle introductions to these topics, though tastes vary and you may prefer other books:

  • Stewart and Tall's Algebraic Number Theory (mentioned in answers to linked questions): treats number fields, with a little bit on elliptic curves and modular forms. gentle, but doesn't cover some important topics.

  • Serre's Course in Arithmetic: p-adics, quadratic forms, modular forms. beautiful. every number theorist should read.

  • Silverman and Tate's Rational Points on Elliptic curves: nice elementary treatment of elliptic curves

  • Kato-Kurokawa-Saito's Number Theory I, II: brief treatments of various topics such as $p$-adics, number fields, zeta functions, elliptic curves. Sequel describes class field theory without proofs.

I'm not saying you should read all of these right now, or in this order, or that reading these will give you deep enough knowledge to do serious research on these topics. I am suggesting you learn the basics of some of the major topics in (algebraic-ish) number theory. (The above list of books still don't cover all the "basics"--e.g., I'm worried Gauss' theory of binary quadratic forms will be missing and Dirichlet's theorem on arithmetic progressions maybe missing, though if you get through the above you can find this stuff in my number theory ii notes, for instance, which also take the approach of a smattering of topics). Then you can try to go deeper in what you're most interested in, or maybe you'll be in grad school with more guidance at that point.

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I can also recommend Neukirch's and Lang's Algebraic Number Theory, Weil's Basic Number Theory, the books here: https://www.mathi.uni-heidelberg.de/~schmidt/publ_en.html#books, Silverman's books on Elliptic Curves and Milne's online notes.

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