ALGEBRAIC FIELD EXTENSIONS
Learning outcomes of the course unit
Good knowledge of the basic aspects of algebraic number theory.
Course contents summary
Integral extensions: algebraic elements, minimal polynomials, primes in integral extensions, "going up" and "going down" theorems, integrally closed domains.
Dedekind domains: noetherian rings, local Dedekind domains, unique factorization of ideals, class group.
Number fields: finite extensions of the rationals, embeddings in the complex numbers, norm and trace maps, discriminant, ring of integers, examples: quadratic, cubic and cyclotomic fields.
Factorization of primes: factorization in rings of integers, ramification index and inertia degree, Kummer's theorem, Dedekind's theorem, factorization and Galois theory, examples: quadratic and cyclotomic fields.
D.A. Marcus "Number Fields" Universitext, Springer-Verlag.
M.R. Murty - J. Esmonde "Problems in Algebraic Number Theory" GTM 190, Springer-Verlag.
J.S. Milne "Algebraic Number Theory" http://www.jmilne.org/math/CourseNotes/ant.html
Assessment methods and criteria