Math 595 Class Field Theory 1 (MWF 10-10:50 pm, AH 347) Instructor: Iwan Duursma, AH 303 Prerequisite: Math 530 Algebraic Number Theory Class field theory is the study of abelian extensions of number fields. Using class field theory many well known properties of quadratic extensions of the rationals generalize to arbitrary finite abelian extensions of number fields. The principal ideal theorem in class field theory establishes an isomorphism between the class group of a number field and the maximal unramified abelian extension of the number field. The main theorems of CFT classify all finite abelian extension of a number field by establishing a bijection, known as Artin reciprocity, with the collection of open subgroups of the idele class group of the number field. In this course we formulate and prove the main results of CFT. Topics to be discussed include: Quadratic forms, Ray class fields, Hilbert class fields, Principal ideal theorem, Tate cohomology groups, Class formations, Artin reciprocity, Idelic formulation, Locally compact groups, Kummer extensions, Existence theorem, Local class field theory, Class field theory for function fields, Artin-Schreier extensions. Good sources for the material of the course are: Milne's course on Class Field Theory (downloadable from his website) and Neukirch's Algebraic Number Theory. Other useful texts: Primes of the form x2 + ny2, by David Cox (good to pick up motivation for CFT), Class Field Theory, by Nancy Childress (Universitext paperback, more accessible than Milne, Neukirch, some drawbacks as well; free download from Springerlink if you connect via UIUC account), Fourier Analysis on Number Fields (GTM 186), by Ramakrishnan and Valenza (different approach). 1Topics course in Algebraic Number Theory. To some extent the chosen topic is flexible and can be changed into Elliptic curves, Cyclotomic fields, Iwasawa theory or other. Contact instructor or share your preferences with one of Dane Skapelund, Nick Anderson, Arindam Roy 1.