COURSE AIMS AND OBJECTIVES: The aim of the course is to present the fundamental results of the algebraic number theory, and to illustrate applications of these results to the last Fermat theorem.
COURSE DESCRIPTION AND SYLLABUS:
1. Traces and norms in field extensions.
2. Discriminant of n-tuple.
3. Computation of discriminant in some cases.
4. Structure of the additive group of the ring of integers in the algebraic number fields.
5. Integral bases
6. Discriminant of an algebraic number field.
7. Ring of integers in the cyclotomic fields, the case of prime power.
8. Ring of integers in the cyclotomic fields, general case.
9. Dedekind's domains and the rings of integers in algebraic number fields.
10. Ideal class group.
11. Factorization in Dedekind'sw domains.
12. Factorization of ideals in extensions of algebraic number fields, examples.
13. Ramification index, inertial degree.
14. The sums of four squares.