General information

• Lecturer: Lauri Tart
• Lectures: 24 hours, Mondays 16:15-18:00, Narva mnt. 18 - 2039
• Tutorials: 8 hours, Mondays 16:15-18:00, Narva mnt. 18 - 2039
• Credits: 3EAP, including 46 hours of self-study

Current

• 27.11 lecture at 16.15-18.00 was virtual (BBB keskonnas).
• Exam dates: 8.01, 16.01, 23.01.

Objectives

The general aim of the course is to give an overview of the fundamentals of the theory of finite fields and teach the basics of finite field arithmetic. The course is meant to cover a number of topics important for applications (especially coding theory and cryptology), including various representations of finite fields, roots of unity and cyclotomic polynomials, the theory of normal bases and methods of constructing irreducible polynomials.

Learning outcomes

After passing the course the students:

1. know about, are able to construct and utilize field extensions and splitting fields of polynomials;
2. are familiar with finite fields, know about and are able to prove fundamental properties of the same, can construct finite fields and compute in them;
3. know and can prove Wedderburns theorem about every finite domain being a field;
4. are familiar with irreducible, minimal, primitive and cyclotomic polynomials and their properties, can prove said properties and use these polynomials to construct more irreducible polynomials and create alternative representations of finite fields;
5. know and can find n-th roots, roots of unity and primitive roots of unity in finite fields;
6. know the trace and norm of an element in a finite field, dual and normal bases of finite fields; are aware and can prove fundamental properties of all those, including the Normal Basis Theorem;
7. know the basics of effective computation in finite fields, including the use of optimal normal bases, can prove the existence of optimal normal bases of types I and II;
8. are familiar with elliptic curves over finite fields, can compute in them and know about the discrete logarithm problem for elliptic curves and its applications.