Best self study books for Algebraic Number Theory?

Question

I have experience in abstract algebra up to fields and field extensions using Artin's Algebra. I am wondering what book would be the most user friendly but also rigorous introduction to algebraic number theory.

Answer 1

Algebraic Number Theory and Fermat's Last Theorem by Stewart and Tall is an amazing introduction into the various modern topics. It revisits the basic concepts of rings and fields and then dives into advanced topics such as class field theory.

For more information, see:

https://www.maa.org/press/maa-reviews/algebraic-number-theory-and-fermats-last-theorem-0

Answer 2

I would recommend Number Fields by Marcus. It is described on that linked web site as follows:

• Contains over 300 exercises

• Assumes only basic abstract algebra

• Covers topics leading up to class field theory

Recently a second edition was published; that is reviewed here.

Answer 3

Algebraic Number Theory by Jarvis has the advantage (for self-study) of having extensive hints and solutions to exercises in an appendix. It's also very accessible - a light knowledge of groups, rings and fields, along with some elementary number theory and linear algebra is plenty. Topics such as ideals and quotient rings are introduced in the text. It's gentler than Marcus or Stewart & Tall, which both require a stronger background in abstract algebra.