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


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.