COURSE AIMS AND OBJECTIVES: The aim of the course is to introduce students to basic notions of the algebraic number theory, and to expose the most basic results of this theory. Algebraic number theory is an important part of the number theory which studies algebraic number fields and the ring of integers in them, which are basic objects of the algebraic number theory (algebraic number field is a finite extension of the field of rational numbers). In the course a particular attention will be paid to applications of algebraic number theory to the last Fermat theorem.
COURSE DESCRIPTION AND SYLLABUS:
1. Review of the basic notions and results from the commutative algebra which will be used in the course.
2. Ring Gauss integers.
3. Pitagorean triples.
4. Arithmetics in the ring of Gauss integers, sums of two squares.
5. A special case of the last Fermat theorem.
6. The last Fermat theorem: preliminary considerations.
7. Solving of the Fermat equation in come cases.
8. Embeddings of the fields.
9. Primitive element theorem, normal extensions.
10. Galois group.
11. Finite fields.
12. Ring of integers in the algebraic number fields.
13. Cyclotomic fields.
14. Kummer's lemma.