Galois theory. Review of basic algebraic notions. Simple extensions, algebraic extensions, separable polynomials, separable extensions and purely inseparable extensions, normal and Galois estensions. Finite fields, fundamental theorem of Galois theory, primitive element theorem, Dedekind theorem about independence of characters, cyclic extensions, Galois theorem about solvability by radicals, fundamental theorem of algebra. Traces, characteristic polynomials and discriminants in algebras.
Number Theory. The ring of algebraic integers of a number field, its structure as an additive group and the discriminant of an algebraic number field. Behaviour of the ring of integers under composition of fields. Computation of relevant ring of integers (of a quadratic fields and of cyclotomic extension). The concept of Dedekind domain, their properties and proof of the fact that the ring of integers of number fields are Dedekind domains. Factorization of primes. Characterization of the ramification of primes by means of the discriminant.
The final examination consists of a written exam and an oral discussion, to be given in the same session. The written exam consists of exercises and questions about the theory (like proving results similar to those that have been seen during the course). It is not allowed to use notes, books or calculators. The students that passed positively the midterm exam (in the part concerning the exercises and/or the theory) have the right to get a one exercise reduction (in the part concerning the exercises and/or the theory) in the written exams of January and Fernuary, not the next sessions.