Algebraic Number Theory (Local fields)

Time and venue: Wednesday, 12-14, MI 03.10.011 

Content: We study the field of p-adic numbers Q_p and its finite extensions. These are called local fields, and are a "local" analogue of number fields which are one of the central objects of interest in algebraic number theory. They are easier to study than number fields, but the theory is to some extent parallel to the global theory, and has many applications to number theory.

One of our main goals is local class field theory that establishes a bijection between finite Galois extensions of a local field K with abelian Galois group and certain subsets of the field K itself. On the way we will study many properties of extensions of local fields such as ramified, unramified and tamely ramified extensions, we will introduce tools such as group cohomology and see many examples.

Books on the subject:

Milne: Class field theory (script online)

Neukirch: Klassenkörpertheorie

Neukirch: Algebraische Zahlentheorie / Algebraic Number Theory

Cassels/Fröhlich (ed.): Algebraic Number Theory

Serre: Corps locaux / Local fields

 

Prerequisites: Algebra 1, 2.

If you plan to attend without having attended a class on Commutative Algebra, please contact me for information on which parts of Commutative Algebra you need as a prerequisite.

Participation in last semester's seminar on Algebraic Number Theory (Prof. Kemper) is helpful, but not required.