Algebraic Number Theory, Part III Michaelmas 2012

Lectures: 9:00-10:00, Tuesdays, Thursdays and Saturdays, MR4

Lecture notes: sections 1-2 - sections 3-4 - sections 5-8 - sections 9-10

First examples class: Monday 29 October, 2:00-3:30, MR3

Second examples class: Tuesday 20 November, 1:00-2:00

Third examples class: Monday 21 January, 2:00-3:30, MR13

Example sheet #3 (revised 15/1/2013) - Solutions

Course description (from the Part III handbook):

In recent years one of the most growing areas of research in number theory has been Arithmetic Algebraic Geometry, in which the techniques of algebraic number theory and abstract algebraic geometry are applied to solve a wide range of deep number-theoretic problems. These include the celebrated proof of Fermat's Last Theorem, the Birch-Swinnerton-Dyer conjectures, the Langlands Programme and the study of special values of L-functions. In this course we will study one half of the picture: Algebraic Number Theory. I will assume some familiarity with the basic ideas of number fields, although these will be reviewed briefly at the beginning of the course. (The relevant algebra will also be found in the Commutative Algebra course.)

Topics likely to be covered (not in order):

• Decomposition of primes in extensions, decomposition and inertia groups. Discriminant and different.
• Completion, adeles and ideles, the idele class group. Application to class group and units.
• Dedekind zeta function, analytic class number formula.
• Class field theory (statements and applications). L-functions.

Prerequisite Mathematics

A first course in number fields (or equivalent reading). Basic algebra up to and including Galois theory is essential.

Literature

• 1. J.W.S. Cassels and A. Frohlich, Algebraic Number Theory. London Mathematical Society 2010 (2nd ed.)
• 2. A. Frohlich, M.J. Taylor, Algebraic Number Theory. Cambridge, 1993.
• 3. J. Neukirch, Algebraic number theory. Springer, 1999.

a.j.scholl@dpmms.cam.ac.uk