The aim of studying and bibliography

The main aim of study: To teach students the basic notions and theorems from Number Theory, Group Theory, Rings and Fields.

The general information about the module: Regular studies, semester III, lectures 30 hours, exercises 30 hours, ends with an exam.

Bibliography required to complete the module

Bibliography used during lectures

1	A. Białynicki-Birula	Algebra	PWN, Warszawa,.	1980
2	J. Browkin,	Wybrane Zagadnienia Algebry	PWN, Warszawa.	1968
3	. B. Gleichgewicht	Algebra	Oficyna Wydawnicz G i S Wrocław.	2002

Basic requirements in category knowledge/skills/social competences

Formal requirements: The student satisfies the formal requirements set out in the study regulations

Basic requirements in category knowledge: Knowledge of linear algebra and basic notions of set theory.

Basic requirements in category skills: performs operations on matrices, computes determinants

Basic requirements in category social competences: Has the ability to define priorities needed for the realization of a particular task, determined by himself/herself or by others.

Module outcomes

MEK	The student who completed the module	Types of classes / teaching methods leading to achieving a given outcome of teaching	Methods of verifying every mentioned outcome of teaching	Relationships with KEK	Relationships with PRK
01	uses the Algorithm of Euclid to solve diophantine equations.	lectures, exercises	written exam and written tests	K_W01+ K_W02+ K_W03+ K_U01+ K_K01+	P6S_KK P6S_UK P6S_WG P6S_WK
02	knows examples of groups, rings and fields.	lectures and exercises	written exam and written tests	K_W05+ K_U05+ K_K01+	P6S_KK P6S_UK P6S_UW P6S_WG
03	checks properties of groups and rings	lectures and exercises	written exam and written tests	K_W04+ K_U17+ K_K01+	P6S_KK P6S_UW P6S_WG P6S_WK
04	knows the notion of an algebraic number and a transcendental number	lectures and exercises	written exam and written tests	K_U01+ K_K01+	P6S_KK P6S_UK

Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).

The syllabus of the module

Sem.	TK	The content	realized in	MEK
3	TK01	Elements of Number Theory. Divisibility of the natural numbers. Common divisor. Factorisation into prime factors. The Algorithm of Euclid. Undetermined (Diophantine) equations. Congruences.	W01 - W03, C01-C03	MEK01
3	TK02	Groups. Examples. Groups of isometries of polygons. Subgroup, coset, Theorem of Lagrange. Normal subgroup, factor-group. Homomorphisms. Cyclic subgroups. Groups of permutations. Finitely generated abelian groups. Groups of matrices.	W04 - W09, C04 - C09	MEK02 MEK03
3	TK03	Rings, ideals, homomorphisms, factor-rings. Zero-divisors, integral domains. Rings of polynomials.	W10 - W 12,C10 - C12	MEK02 MEK03
3	TK04	Fields, field extension. Algebraic elements. Algebrically closed field.	W13 - W15 ,C13 - C15	MEK04

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 3)	The preparation for a test: 10.00 hours/sem.	contact hours: 30.00 hours/sem.	complementing/reading through notes: 15.00 hours/sem. Studying the recommended bibliography: 5.00 hours/sem.
Class (sem. 3)	The preparation for a Class: 10.00 hours/sem. The preparation for a test: 8.00 hours/sem. Others: 8.00 hours/sem.	contact hours: 30.00 hours/sem.	Finishing/Studying tasks: 7.00 hours/sem.
Advice (sem. 3)	The preparation for Advice: 3.00 hours/sem.	The participation in Advice: 3.00 hours/sem.
Exam (sem. 3)	The preparation for an Exam: 20.00 hours/sem.	The written exam: 2.00 hours/sem.

The way of giving the component module grades and the final grade

The type of classes	The way of giving the final grade
Lecture	No grade for participation in the lecture.
Class	Passing grade from exercises based on grades from two tests. In the border case the activity during the execises may tip the scale.
The final grade	Final grade based on the final exam. Passing grade form the exercises is the condition to take the exam. In the border case good grade from the exercises or a few oral questions may tip the scale by one half of a unit (say from 4 to 4,5).

Sample problems

Required during the exam/when receiving the credit
egzamAlgebra2010.pdf
Egzamin3Alg2012.pdf

Realized during classes/laboratories/projects
koloAlg2011.pdf
koloIIAlg2011.pdf

Others
(-)

Can a student use any teaching aids during the exam/when receiving the credit : no

The contents of the module are associated with the research profile: yes

1

D. Wajnryb

The braid group and its presentation

2021