Algebraic Number Theory

Time: the second semester

Course Classification: compulsory course

Credit: 3 hours

Period: 54 hours

Previous courses: Abstract Algebra

Syllabus:

(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, Algebraic Number Theory, Thompson Publishing Co., 1967.

(2) S. Lang, Algebraic Number Theory, Springer-Verlag, GTM 110.

(3) J. S. Milne, Algebraic Number Theory, available at http://www.jmilne.org/math/.

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