Algebra 2 (NMAI076) - Spring term 2025

Lecture: Thursday 12:20 - 13:50, lecture room S8
Exercises: Thursdays 14:00 - 15:30 on odd weeks, lecture room S1

Lecture notes	Czech	English
Algebra 1	alg1_cz	alg1_en
Algebra 2	alg2_cz	alg2_en

Evaluation:
To obtain the credit (Zápočet) for the course you need to score at least 54 out of 90 points. Points can be obtained from 3 homework assigments (3*30 points). The final grade will be determined by an oral exam, admission to the exam requires getting ''Zápočet'' first.

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
20/02	Group homomorphisms and isomorphisms, invariants, classifications Ex: Examples	Section 1
27/02	Quotient groups Homomorphism theorem and isomorphism theorems	Section 2
06/03	Ideals, Principal Ideal Domains Ex: Quotient groups, sums and intersections of Ideals	Section 3	HW1 due 20.03.
13/03	Ring homomorphisms, quotient rings Isomorphism theorems for rings, prime/maximal ideals	Section 4
20/03	Ring and field extensions Ex: quotient rings	Section 5
27/03	Cancelled due to illness
03/04	Algebraic numbers, minimal polynomials, degrees of simple/multiple extensions Ex: Examples, constructibility in classic geometry	Section 6, 7	HW2 due 17.04.
10/04	Rupture and Splitting fields Frobenius endomorphism	Section 8, 9.1, 9.2
17/04	Classification of finite fields; modular representations, DFT Ex: primitive roots of unity	Section 9.3, 10
24/04	Fast Fourier Transform (FFT), Fast polynomial multiplication The ring of power series	Section 10,11	HW3 due 15.05.
01/05	International Workers' Day
08/05	V day
15/05	Fast polynomial division, Berlekamp's algorithm Ex: square-free factorization	Section 11,12
22/05	General algebraic structures	Section 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.