# 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
13/02Group homomorphisms and isomorphisms, invariants, classifications
Ex: Examples
Section 1
20/02Normal subgroups, quotient groups, homomophism theorem,
1st isomorphism theorem
Section 2
27/02Ideals and divisibility, PIDs, ideals in fields
Ex: quotient groups, intersection and sum of ideals
Section 3
06/03Quotient 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, 51st HW
due 27.3
20/03Algebraic and transcendental numbers
Minimal polynomials of algebraic numbers
Section 6
27/03Extensions 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
10/04Easter Monday
17/04Modular representations of rings, Fast Fourier TransformationSection 10
24/04Fast polynomial multiplication and division, formal power series
Ex: general field extensions and their degree
Section 11
01/05International Workers' Day
08/05V-Day2nd HW
due 24.5
15/05Square-free factorizations of polynomials
Berlekamp's algorithm for decomposition into irreducibles
Section 12
22/05Berlekamp'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.