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I went through this thread on this question asking about a text on number theory, but my question is slightly different.

I was looking for a short and sweet book on elementary number theory. I don't need it to be a definitive treatise on number theory. I just need to have book which covers enough material to advance to algebraic number theory. I have looked at the books by Niven-Zuckerman-Montgomery and the Jones' and the one by Burton. All of them were very readable and I would like to read at least one of them at some point. For now, I would just like a book (say 100 pages long) which can get me acquainted with most of the material needed for algebraic number theory. Basically there is a course on algebraic number theory that I wish to take which begins in a month's time and I have not done enough elementary number theory.

Community
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Derek Scavo
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3 Answers3

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The thing about algebraic number theory is that, although the problems you will be looking at come from "classical" number theory, the techniques owe a lot more to abstract algebra than to classical number theory. You would be better served making sure that you have a solid understanding of commutative rings, field extensions, Galois Theory, and the basics of modules over commutative rings.

In that respect, the book by Ireland and Rosen that was mentioned in that thread, A Classical Introduction to Modern Number Theory, may be a good one, since it is leading you in that direction in any case.

When I took Algebraic Number Theory (which I absolutely loved; it was a graduate two-course sequence), I had not had any number theory beyond the very elementary congruence stuff; I had never seen anything like Quadratic Reciprocity, or Fermat's Christmas Theorem, or anything like that. I did not feel hobbled by this lack. But I certainly had to lean heavily on the commutative algebra and Galois Theory I knew!

It was only later that I sat in on an upper-division number theory course (based on the Ireland-Rosen book mentioned above), and that I read through Gauss's Disquisitiones (on my own) and other number theory books. Then again, I'm not a number theorist, so perhaps I am in fact deeply handicapped by this but I don't know it.

Arturo Magidin
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  • Thanks. I feel I have a good background in algebra. It's comforting to hear that not knowing too much elementary number theory won't peg me down. I have read about elementary divisibility, Chinese remainder theorem, Fermat's little theorem, Wilson's theorem, etc and I am comfortable with these. I have no idea about quadratic reciprocity and continued fractions. – Derek Scavo Nov 25 '10 at 06:07
  • @Derek Scavo: That's about where I was, number-theory-wise. – Arturo Magidin Nov 25 '10 at 06:46
  • +1 I think this advice is spot on. You don't need to read any elementary number theory before learning algebraic number theory. Knowing the statement of quadratic reciprocity helps, but that's about it. – Alex B. Nov 25 '10 at 08:33
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The book of Harold Davenport, The Higher Arithmetic: An introduction to the theory of numbers, fits your bill.

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Elementary Theory of Numbers by Sierpinski.

Matt Calhoun
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  • Thanks, Matt. Although that looks fatter than most of the other books I have seen. – Derek Scavo Nov 25 '10 at 06:07
  • You don't have to read the whole thing at once you know =) This book is one of my all time favorites, I can almost guarantee you will love this book if you put a serious effort into it. – Matt Calhoun Nov 25 '10 at 23:22
  • Yes. But it's difficult to decide what to skip, which was the reason for this question. – Derek Scavo Nov 26 '10 at 02:53