## Algebra 2 (NMAI076) - Spring term 2024

Lecture: Thursday 14:00 - 15:30, lecture room S6
Exercises: Thursdays 15:40 - 17:10 on odd weeks, lecture room S6

Lecture notes Czech English
Algebra 1 alg1_cz alg1_en
Algebra 2 alg1_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
22/02Repetition groups, group homomorphisms and isomorphisms, invariants
Ex: Examples
Section 1.1-1.3
29/02 Group classifications; Normal subgroups, quotient groups Section 1.4, 2
06/03 The homomomorphism theorem and isomorphism theorems for groups; Ideals, PIDs
Ex: determining quotient groups, sums and intersections of ideas in Z
Section 2, 3 HW1
due 21/03
13/03 Ring homomorphisms, quotient rings Section 4
20/03 Isomorphism theorems for rings, prime/maximal ideals, ring and field extensions
Ex:computing quotient rings, ring extensions
Section 4,5.1
28/03 Algebraic and transcendental elements, the degree of a field extension, minimal polynomials Section 5.2,6.1,6.2
04/04 minimal polynomials, degrees of general extensions, algebraic numbers are a field
Ex: computing minimal polynomials, degrees of extensions, splitting fields
Section 6.2, 6.3 HW2
due 18/04
11/04Problems solvable by ruler and compass
Uniqueness of rupture fields and splitting fields
Section 7, 8
18/04The classification of finite fields
Ex: Constructability of regular n-gons
Section 9
25/04Modular representations, Fast Fourier Transform Section 10
02/05 (self study) fast polynomial multiplication and division using FFT
Ex: primitive roots, FFT
Section 11 HW3
due 16/05
09/05fast polynomial multiplication and division using FFT
square-free decomposition
Section 11, 12.1
16/05 Berlekamp's algorithm
Ex: formal power series, discussion of 3rd homework
Section 12
23/05

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.