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 andL-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., 1967.

(2) L. J. Goldstein, Analytic Number Theory, Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1971.

(3) S. Lang, Algebraic Number Theory, Springer-Verlag, GTM110.

(4) J. S. Milne, Algebraic Number Theory, available athttp://www.jmilne.org/math/

(5) H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambrige University Press, 2001.

(6) A . Weil, Basic Number Theory, Springer-Verlag.