Undergraduate Number Theory Book Recommendations

[daniele1234]

Hi , everyone! This is my first post/thread/anything on this forum so first I apologise if I slip up or make any mistakes. Anyway, my question is about recommendations for textbooks for Undergraduate Number Theory. So far, I have studied Calculus 1-3 (including things like line integrals, Stoke's Theorem, divergence theorem, etc.) abstract algebra (undergraduate abstract algebra Serge Lang), some analysis and linear algebra. I have not really done much number theory before but know a few of the more obvious results (things like Chinese Remainder Theorem). I am actually planning to self-teach myself Number Theory so I am looking for a book ideally that has plenty of examples and exercises to cement my knowledge. So, do any of you have any recommendations or advice?

Thank you

(Also do you think given what I know so far I am prepared to tackle Number Theory?)

 

[The Bill]

Kenneth H. Rosen's Elementary Number Theory is a decent low level survey and introduction. It's nothing special, but it's serviceable, and the 5th edition and before are cheap used.

 

[daniele1234]

Anyone else? (Thank you The Bill for the one above - how advanced would you say it is and would you say it is advanced enough for undergraduate level?)

 

[The Bill]

Rosen's book is aimed at undergraduate math students. At the university I went to, a class based on it would probably be mostly attended by second through fourth year students as an in-major elective. It has minimal prerequisites. If you're studying university level maths at all, you're ready for it.

 

[Wrichik Basu]

Problem Solving Strategies by Arthur Engel, Springer Publication.

The book doesn't have much of theory but some of world's unsolved sums.

 

[MidgetDwarf]

Since you have studied abstract algebra from Serge Lang. You may be able to forget about Rosen, and try a more challenging book. Maybe give Apostols number theory book a go.

You can always supplement it with Rosen if it is too hard.

 

[daniele1234]

@MidgetDwarf Thank you very much for your response and from what I have read the Apostles book is rated very highly. I do notice however that it is only analytic number theory. I was hoping for a number theory book that was both algebraic and analytic. Do you think I should buy a number theory book with both algebraic and analytic in one, or two separate books one for analytic number theory (e.g. the Apostols) and algebraic number theory (e.g. Lang)?

 

[mathwonk]

for starters i would suggest one more elementary than either algebraic or analytic number theory, say Elementary Number theory by Vanden Eynden. the point is that this book has few prerequisites.

Algebraic number theory, say by Neukirch, tends to assume you know already Galois theory for example. and analytic number theory assumes complex analysis. So I think you seem ready for the more elementary approach. And I recommend it.

after glancing at the book by apostol. it also seems excellent. i have found his writing vastly superior to that of Lang in general, i.e. more precise, correct, and understandable, although occasionanlly Lang is inspiring and useful. but apostol always considers the needs of his student audience and works hard to eliminate errors. Lang is brilliant but often seems just to wing it.

a quick look at langs book reveals a high level of sophisticated algebraic prerequisites, such as local rings and dedekind domains, making it probably suitable for a second year graduate course.

 

[Infrared]

For algebraic number theory, I'd recommend James Milne's notes on the subject, available freely online. You can find them by googling.