Number Theory Texts: Suggestions & Prerequisites for Undergraduates

[a7d07c8114]

A senior friend of mine who is going to graduate school in mathematics suggested that I try to get at least some exposure to number theory before applying to/attending graduate school. (I'm a freshman undergrad.) Well, I was going to do so anyway, since it's interesting and even applicable, but now I need some direction. My university offers an elementary introduction to number theory at the undergraduate level. My friend has informed me that with some abstract algebra, the entire course will be trivial.

I'd like to see what options there are for studying number theory in terms of textbooks, so I can maybe arrange some independent study with a professor or grad student. My aforementioned friend has suggested Rosen & Ireland's "A Classical Introduction to Modern Number Theory." Please suggest any books you have found useful, and please include the prerequisite background in mathematics/other sciences appropriate for each suggestion. Suggestions don't have to require any particular background knowledge; just fire away, as long as you include what's required to really benefit from the text.

Thanks in advance to all responders!

 

[deluks917]

"A classical intro to Moden Numer Theory" is a fairly technical intro book imo. I think its very good but maybe look at it at a library or something before getting it if you are a freshmen. Also it presupposes some abstract algebra.

 

[Infinitum]

For basics, I liked Elementary Number Theory by David Burton. Gives you a good idea of how things go about in number theory, and just high school knowledge of math will be sufficient to get started.

 

[mathwonk]

i wouldn't necessarily take your friend's dismissal of the university course as gospel either. maybe its because i am a professor, but none of my courses is ever trivial, in my opinion, and i would be surprised if that is true at your school either.

last time i taught number Theory we spent the first day just writing down as many primes as we could and looking at the list, pretty trivial, and ended with dirichlet's proof of the existence of infinitely many primes in (suitable) arithmetic progressions, not so trivial.

it is also usually much easier and more efficient to learn from someone who is taking the trouble to explain to you what he knows than to read on your own. why not take advantage of that?

oh, another nice elementary book is the one by van den eynden. my favorite less elementary book is by trygve nagel. most number theory books have few prerequisites. that's why people like number theory. the only exception i know of is Basic number theory, by Andre Weil, which assumes Haar measure on page one.Whatever book you choose, apparently the main point of an elementary course, is to master quadratic reciprocity.

By the way, you might as well take a look at Gauss' Disquisitiones Arithmeticae.

 

[a7d07c8114]

Thanks to all those who replied. mathwonk, perhaps I put my friend's words in too negative a light, which I would never want to do. What he said, more precisely, was that it would be nice to see number theory in some form, but number theory is more or less optional before graduate school whereas abstract algebra is of course mandatory; and since I would be learning abstract algebra anyway, it may be better to hold off on number theory, since apparently a lot of elementary number theory appears obvious once you have an understanding in abstract algebra. I don't believe he was completely dismissing the number theory course here, just throwing a few suggestions out there (and emphasizing that many professors are more than willing to set up some sort of independent study).

I'll keep this page bookmarked for now and come back to it when I begin number theory proper. Thanks all!