# Algebraic Number Theory

Prerequisites: Solid knowledge of undergraduate algebra including galois theory and module theory over PID and some basic commutative algebra at the level of atiyah&mcdonald.

If you are interested, please message hochs. The seminar will start when we gather enough interested people.

List so far: hochs, HBT, vfr45678i

List of references:

Algebraic Number Theory - Cassels and Fröhlich (*http://amzn.com/0950273422*)

Algebraic Number Theory - Jürgen Neukirch (*http://amzn.com/3540653996*)

Algebraic Number Theory - Lang (*http://amzn.com/0387942254*)

Local Fields - Serre (*http://amzn.com/0387904247*)

Class Field Theory - Artin and Tate (*http://amzn.com/0821844261*)

Pros and Cons of the books above:

Cassels & Frohlich is a classic with the approach to CFT via group cohomology, covering both local and global class field theory (same as Serre). It also has Zeta-Functions and L-functions, as well as a treatment of semi-simple algebraic groups & Tate's original Fourier Analysis thesis. It doesn't, however, have very in-depth treatment of computations with group cohomology - it builds just the bare minimum for the statements of CFT to fall out.

Serre's Local Fields has much more in the way of group cohomology / brauer groups, e.g. techniques for computing the local symbols. It also has treatments of Witt Vectors and non-abelian cohomology.

Neukirch's ANT book is the most user-friendly, with lots of motivating examples and exercises. However, its treatment of completions of discriminants, differents, and compactness of adele/idele theories in the later chapters seem unnecessarily technical and nontransparent. This is a good introductory book for someone who wants lots of examples and down-to-earth treatments of ANT.

Lang's ANT does everything via Ray Class Groups, where you need to keep track of field extension arithmetics (not very convenient). It is also more analytical, and gets to global class field theory the quickest, but it proves weaker versions of CFT. It, however, has sections on Tauberian Theorems.

Artin and Tate's CFT is a very complete account of global class field theory. I like this since it has a thorough treatment of Grunwald-Wang Theorem and local symbol computations.