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I just finished an undergraduate course on elementary number theory. I covered the following topics:

  • Divisibility, GCD, Euclidean Algorithm, Linear Diophantine Equations
  • Primes, Fundamental Theorem of Arithmetic
  • Congruences, CRT, Fermat's Little theorem, Wilson's theorem
  • Number Theoretic Functions
  • Euler Totient Function and Euler's generalization of FLT
  • Primitive Roots, Indices
  • Quadratic Residues, Euler's Criterion, Quadratic Reciprocity law

I wish to learn more number theory, But I don't know how to proceed. I've heard about Analytical and Algebraic Number Theory, but I don't know what that is and how/when I should study it.

So I guess my question is:

  • What should I learn next if my goal is to learn more number theory?
  • What books should I refer to?
  • Are there any other fields I should focus on that may help me in number theory?

EDIT Not only asking for books on number theory, but also what other subjects I should study, in what order I should study etc.

Edmaand Vlad Phiyik
  • 185

1 Answers1

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If you had only elementary number theory as a course, then you probably also want to read some basic abstract algebra before going further. The book "Notes from the Underground" by Aluffi is fairly readable. However you do not need to read everything in that book and Aluffi takes a somewhat different approach, as he talks earlier then normal about modules.

Besides that you can also just try to read some book on Algebraic Number Theory.

Maybe the book by Frazer Jarvis is suitable for your needs. It does not assume much. At least in my opinion. It also has a short chapter on "Analytic Methods".

Edit: Different books that come to mind:

  • Fermat's Dream, by Kato and others.

As I understand it, it is more of a "What is Algebraic Number Theory about". I do not think you really learn the topic from that book.

  • Algebraic Number Theory for Beginners, by John Stillwell.

As far as I can judge it is a pretty down to earth introduction to Algebraic Number Theory, but as the book above I would not say you really learn the topic.

  • A Conversational Introduction to Algebraic Number Theory, by Paul Pollack.

Never really looked into it. I think the title says it all...

These are some "different" kind of textbooks. The book by Jarvis is a real textbook but still a gentle one.

MathUnicorn
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