Time: the third semester.
Credit: 3 hours.
Period: 54 hours.
Previous courses: Abstract Algebra

Course contents:

(1) Numbers and Ideals: the ring of integers, ideals and factorization, embedding      in the complex numbers, change of fields;
(2) Valuations: valuations and completions, field extensions and ramification, the      different, Ideles and Adeles;
(3) Special fields: quadratic fields, pure cubic fields, biquadratic fields, cyclotomic       fields, class numbers of cyclotomic fields, Fermat’s Last Theorem;
(4) Analytic methods: zeta functions and L-series, analytic continuation and the      functional equation, density theorems;
(5) Class field theory: the classical theory, chevalley’s reformulation, reciprocity       theorems, the Kronecker-Weber Theorem.

References:

(1) J. W. S. Cassels and A. Frohlich, eds., Algebraic Number Theory, Thompson
      Publishing Co.
(2) L. J. Goldstein, Analytic Number Theory, Prentice-Hall, Inc. Englewood Cliffs,
      New Jersey.
(3) S. Lang, Algebraic Number Theory, Springer-Verlag, GTM110;
(4) H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cam-
     brige University Press.
(5) A . Weil, Basic Number Theory, Springer-Verlag.