Math 676: Algebraic Number Theory — Fall 2020

1. The book Algebraic Number Theory by Frohlich and Taylor is perhaps the closest to this course in terms of choice of material.
2. Milne's Algebraic Number Theory follows the same general outline as this course.
3. The book Algebraic Number Theory, edited by Cassels and Frohlich, is a classic, with articles by top experts which together cover a great deal of material, of which only a small fraction is covered in this course.
4. Lang's book Algebraic Number Theory is a standard reference. Like many of Lang's books, in hindsight it does everything the right way, but this isn't always clear when learning the subject for the first time.
5. Koch's book Number Theory: Algebraic Numbers and Functions is very nice, and organizes the material in a different order than this course, doing geometry of numbers first before developing the theory of ideals. It also has a good deal of information about L-series.
6. Neukirch's book Algebraic Number Theory takes a modern or even futuristic approach to the subject, developing the material with an eye towards the perspective of current research in Arakelov theory. The approach in this book is more sophisticated than that in any other, which will appeal to some readers based on their specific background and taste.
7. Artin's book Algebraic Numbers and Algebraic Functions develops the theory of number fields and function fields in parallel, in terms of absolute values rather than ideals.
8. Lorenzini's book An Invitation to Arithmetic Geometry develops the theory of number fields and algebraic curves in parallel, emphasizing the connections between them.
9. Koch's book Algebraic Number Theory (previously published as Number Theory II in the book series edited by Parshin and Shafarevich) provides a survey of a massive amount of material, which helps one figure out what's known.
10. Cox's book Primes of the Form x²+ny² makes for outstanding supplementary reading, since it explains the historical context in which various concepts were developed. As a bonus, it is hands-down the best reference I know for providing intuition about class field theory, which is the topic of the follow-up course Math 776.
11. Cassels's book Local Fields gives an excellent treatment of p-adics (especially Newton polygons), with many interesting applications to recursive sequences and other topics.
12. Serre's book Local Fields gives a great treatment of higher ramification groups and many other things.
13. The Silverman-Tate book Rational Points on Elliptic Curves does a masterful job of giving a glimpse of the very important subject of elliptic curves while allowing the reader to get through the book in a day or two.
14. At a more advanced level, Lang's book Survey on Diophantine Geometry (which was previously published as Number Theory III, in a book series edited by Parshin and Shafarevich) surveys a massive amount of advanced material in number theory. This is a great reference to browse for inspiration, and to help one get a sense of the directions of current research.