Algebra 2 (NMAI076) - Spring term 2023

Lecture: Monday 09:00 - 10:30, lecture room S6
Exercises: Monday 10:40 - 12:10 on odd weeks, lecture room S6, held by Filippo Spaggiari

Lecture notes	Czech	English
Algebra 1	alg1_cz	alg1_en
Algebra 2	alg1_cz	alg2_en

Evaluation:
To get ''Zápočet'' (i.e. to pass the exercise classes) you need to score at least 45 out of 70 points. Points can be obtained from 2 homework assigments (2*30 points), and in-class presentation (10 points). For more details see Filippo Spaggiari' website.
The final grade will be determined by an oral exam, admission to the exam requires passing the exercise class.

Syllabus: This course is a continuation of Algebra 1 and aims to introduce computer science students to some topics from abstract algebra:

1. Homomorphisms (group homomorphism, quotient groups, ring homomorphisms, ideals, classification of finite fields)
2. Number fields (ring and field extensions, algebraic elements, and finite degree extensions)
3. Algorithms in polynomial arithmetic (fast polynomial multiplication and division, decomposition)
4. Other algebraic structures (lattices and Boolean algebras)

Overview:

Date	Topics	Lecture notes	Homework
13/02	Group homomorphisms and isomorphisms, invariants, classifications Ex: Examples	Section 1
20/02	Normal subgroups, quotient groups, homomophism theorem, 1st isomorphism theorem	Section 2
27/02	Ideals and divisibility, PIDs, ideals in fields Ex: quotient groups, intersection and sum of ideals	Section 3
06/03	Quotient rings, homomorphism theorem, 1st and 2nd isomorphism theorem for rings	Section 4.1,4.2
13/03	prime and maximal ideals; Ring and field extensions, degree Ex:quotient rings, ring and field extensions	Sections 4.3, 5	1st HW due 27.3
20/03	Algebraic and transcendental numbers Minimal polynomials of algebraic numbers	Section 6
27/03	Extensions by more than one element, constructability, ruler-and-circle constructions Ex: minimal polynomials and algebraic numbers	Section 6.3, 7
03/04	Uniqueness of splitting fields (up to isomorphism), Classification of finite fields	Sections 8, 9
10/04	Easter Monday
17/04	Modular representations of rings, Fast Fourier Transformation	Section 10
24/04	Fast polynomial multiplication and division, formal power series Ex: general field extensions and their degree	Section 11
01/05	International Workers' Day
08/05	V-Day		2nd HW due 24.5
15/05	Square-free factorizations of polynomials Berlekamp's algorithm for decomposition into irreducibles	Section 12
22/05	Berlekamp's algorithm; algebraic structures, substructures, homomorphisms Ex: ordered sets, lattices and Boolean algebras	Sections 12, 13

Consultation:
If you have questions, do not hesitate to ask (either in person or via e-mail)! I have no official office hours, but if required, a personal meeting can be arranged. Please make also use of the exercise classes to discuss your questions.