I’m a freshman math major student and was looking for some recommendations on number theory books. Just to clarify, I was reading Abstract Algebra: An Introduction by Thomas Hungerford.
Although it was my first abstract algebra book, it gave me some background in group and ring theory, modular arithmetic, fields, etc. I was undecided about two books: A Classical Introduction to Modern Number Theory by Ireland and Rosen, and Algebraic Number Theory by Jarvis, but I would like to use only one of them.
At first glance, it seemed that the first one is way more rigorous than the second — specially because it says graduate textbooks, not undergraduate textbooks on it — but it turns out that a lot of people said that this is actually not true, and that the algebraic number theory book by Jarvis is more rigorous.
Because of that, I would like to hear from you opinions on these two books, and I would really appreciate if you could list the prerequisites for each of them. Do I need to know anything besides the basics of abstract algebra to attack these books? Do I need to know some linear algebra, or some analysis or even calculus to handle it?
Moreover, which would be more appropriate — in terms of material covered, clarity and rigor — considering that I have studied abstract algebra using Hungerford's book and that I have a basic (I mean, BASIC) background in elementary number theory? (*) Is there one of them that is more algebraic than the other? Does one of them presents analytic number theory and requires some background in analysis?
I also accept recommendations regarding other books, though I'm really looking for opinions on those two books I mentioned.
(*) I have never used any number theory book before, but Hungerford's book gave me some insights on elementary number theory, and I have attended some college classes in elementary number theory while on my high school senior year.
Best Answer
Ireland and Rosen has a lot of great material in it, but it is not a book from which you should learn algebraic number theory: its treatment of that topic only starts near the middle and I felt that I could follow that part only because I had learned algebraic number theory elsewhere already.
The two books you mention are largely on different aspects of number theory, so I think the idea that you can only use one of them is misguided. Read both!