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Algebra and Number Theory

General data

Course ID:	1100-EA0ENG
Erasmus code / ISCED:	(unknown) / (0610) Information and Communication Technologies (ICTs), not further defined The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Algebra and Number Theory
Name in Polish:	Algebra and Number Theory
Organizational unit:	Faculty of Mathematics and Computer Science
Course groups:
ECTS credit allocation (and other scores):	0 OR 6.00 (differs over time) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
(in Polish) Forma zaliczenia:	(in Polish) egzamin
Level:	BA
(in Polish) Forma studiów:	(in Polish) stacjonarne
Prerequisits:	Knowledge of English.
Short description:	The goal of the course is to present those notions of number theory and abstract algebra which are necessary for the understanding of the modern applications of those branches of mathematics in computer science, e.g. in cryptography.
Learning outcomes:	On completion of the course, the student: LO1. distinguishes between basic algebraic structures, solves simple exercises concerning algebraic structures, LO2. applies modular arithmetic, LO3. solves linear congruences in ℤ and the equations corresponding to them in the ring ℤₙ using extended Euclidean algorithm, LO4. applies matrix calculus, calculates determinants, LO5. solves systems of linear equations, LO6. formulates and interprets the definitions and theorems given in the lecture (for example: Euler’s theorem, Chinese remainder theorem, Cramer’s theorem,...), LO7. has a critical attitude to solutions of the problems. These learning outcomes achieved during the course allow to fulfill the directional effects of education which have the following markings in the programme of Computer Science-Bachelor`s Degree: 1100I-1A_W01, 1100I-1A_W02, 1100I-1A_W03, 1100I-1A_W05, 1100M-1A_U01, 1100M-1A_U02, 1100M-1A_U03, 1100M-1A_U06, 1100M-1A_U07, 1100M-1A_U11, 1100M-1A_U19, 1100M-1A_U21, 1100M-1A_K01, 1100M-1A_K02, 1100M-1A_K05.

Classes in period "Winter Semester 2023/2024" (past)

Time span:	2023-10-01 - 2024-02-25	Selected timetable range: weekly course term Navigate to timetable MO TU W W CK TH FR
Type of class:	Discussion practice class, 28 hours more information Lecture, 28 hours more information
Coordinators:	Szymon Brzostowski
Group instructors:	Szymon Brzostowski, Kacper Grzelakowski
Students list:	(inaccessible to you)
Examination:	Course - (in Polish) Ocena zgodna z regulaminem studiów Discussion practice class - (in Polish) Ocena zgodna z regulaminem studiów Lecture - (in Polish) Ocena zgodna z regulaminem studiów
Teaching Method:	lecture, talk, discussion, brainstorming
Method and Criteria of Assessment:	The class grade is the average of the grades from several tests. It may be increased in special cases (to students taking active part in the classes). The lecture grade is the final (written) exam grade. A passing grade for the class is a prerequisite for taking the theory exam. The final grade is the average of the class grade (50%) and the lecture grade (50%).
Course Content:	1. Groups, rings, fields. Operations on complex numbers. 2. Divisibility. Greatest common divisor. Euclidean algorithm and Bezout's theorem. 3. Modular arithmetic, congruences. Linear Diophantine equations. Prime numbers. 4. Matrices, operations in the set of matrices with coefficients in a field. 5. Systems of linear equations. Gauss-Jordan elimination. 6. Determinants. 7. Finding inverse matrix. 8. Cramer's rule.
Bibliography:	[Gar07] Garrett, P. Abstract Algebra. Available at: http://www.math.umn.edu/~garrett/m/algebra. [Goo06] Goodman, F., M. Algebra: Abstract and Concrete. Available at: http://homepage.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html. [Sho08] Shoup, V. A Computational Introduction to Number Theory and Algebra. Available at: http://shoup.net/ntb.

Classes in period "Winter Semester 2022/2023" (past)

Time span:	2022-10-01 - 2023-02-19	Selected timetable range: weekly course term Navigate to timetable MO CK CK TU W W TH CK CK CK CK FR CK CK CK
Type of class:	Discussion practice class, 28 hours more information Lecture, 28 hours more information
Coordinators:	Szymon Brzostowski
Group instructors:	Szymon Brzostowski, Kacper Grzelakowski
Students list:	(inaccessible to you)
Examination:	Course - (in Polish) Ocena zgodna z regulaminem studiów Discussion practice class - (in Polish) Ocena zgodna z regulaminem studiów Lecture - (in Polish) Ocena zgodna z regulaminem studiów
(in Polish) Czy IRK BWZ?:	(in Polish) T
Teaching Method:	lecture, talk, discussion, brainstorming
Method and Criteria of Assessment:	The exercise class mark is the average of the marks from two tests. It may be increased in special cases (to students taking active part in the exercises) up to one level up. The lecture mark is the final (written) exam mark. A passing mark for exercise class is a prerequisite for taking the theory exam. The final mark is comprised of the class mark (50%) and the lecture mark (50%).
Course Content:	1. Groups, rings, fields. Operations on complex numbers. 2. Divisibility. Greatest common divisor. Euclidean algorithm and Bezout's theorem. 3. Modular arithmetic, congruences. Linear Diophantine equations. Prime numbers. 4. Matrices, operations in the set of matrices with coefficients in a field. 5. Systems of linear equations. Gauss-Jordan elimination. 6. Determinants. 7. Finding inverse matrix. 8. Cramer's rule.
Bibliography:	[Gar07] Garrett, P. Abstract Algebra. Available at: http://www.math.umn.edu/~garrett/m/algebra. [Goo06] Goodman, F., M. Algebra: Abstract and Concrete. Available at: http://homepage.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html. [Sho08] Shoup, V. A Computational Introduction to Number Theory and Algebra. Available at: http://shoup.net/ntb.

Classes in period "Winter Semester 2021/2022" (past)

Time span:	2021-10-01 - 2022-01-23	Selected timetable range: weekly course term Navigate to timetable MO W TU W CK CK CK CK TH FR CK
Type of class:	Discussion practice class, 28 hours more information Lecture, 28 hours more information
Coordinators:	Szymon Brzostowski
Group instructors:	Szymon Brzostowski, Kacper Grzelakowski
Students list:	(inaccessible to you)
Examination:	Course - (in Polish) Ocena zgodna z regulaminem studiów Discussion practice class - (in Polish) Ocena zgodna z regulaminem studiów Lecture - (in Polish) Ocena zgodna z regulaminem studiów
(in Polish) Czy IRK BWZ?:	(in Polish) T
Teaching Method:	lecture, talk, discussion, brainstorming
Method and Criteria of Assessment:	The exercise class mark is the average of the marks from two tests. It may be increased in special cases (to students taking active part in the exercises) up to one level up. The lecture mark is the final (written) exam mark. A passing mark for exercise class is a prerequisite for taking the theory exam. The final mark is comprised of the class mark (50%) and the lecture mark (50%).
Course Content:	1. Groups, rings, fields. Operations on complex numbers. 2. Divisibility. Greatest common divisor. Euclidean algorithm and Bezout's theorem. 3. Modular arithmetic, congruences. Linear Diophantine equations. Prime numbers. 4. Matrices, operations in the set of matrices with coefficients in a field. 5. Systems of linear equations. Gauss-Jordan elimination. 6. Determinants. 7. Finding inverse matrix. 8. Cramer's rule.
Bibliography:	[Gar07] Garrett, P. Abstract Algebra. Available at: http://www.math.umn.edu/~garrett/m/algebra. [Goo06] Goodman, F., M. Algebra: Abstract and Concrete. Available at: http://homepage.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html. [Sho08] Shoup, V. A Computational Introduction to Number Theory and Algebra. Available at: http://shoup.net/ntb.

Classes in period "Winter Semester 2020/2021" (past)

Time span:	2020-10-01 - 2021-02-07	Selected timetable range: weekly course term Navigate to timetable MO TU W CK W CK TH CK FR
Type of class:	Discussion practice class, 28 hours more information Lecture, 28 hours more information
Coordinators:	Szymon Brzostowski
Group instructors:	Szymon Brzostowski, Kacper Grzelakowski
Students list:	(inaccessible to you)
Examination:	Course - (in Polish) Ocena zgodna z regulaminem studiów Discussion practice class - (in Polish) Ocena zgodna z regulaminem studiów Lecture - (in Polish) Ocena zgodna z regulaminem studiów
(in Polish) Czy IRK BWZ?:	(in Polish) T
Teaching Method:	lecture, talk, discussion, brainstorming
Method and Criteria of Assessment:	The exercise class mark is the average of the marks from two tests. It may be increased in special cases (to students taking active part in the exercises) up to one level up. The lecture mark is the final (written) exam mark. A passing mark for exercise class is a prerequisite for taking the theory exam. The final mark is comprised of the class mark (50%) and the lecture mark (50%).
Course Content:	1. Groups, rings, fields. Operations on complex numbers. 2. Divisibility. Greatest common divisor. Euclidean algorithm and Bezout's theorem. 3. Modular arithmetic, congruences. Linear Diophantine equations. Prime numbers. 4. Matrices, operations in the set of matrices with coefficients in a field. 5. Systems of linear equations. Gauss-Jordan elimination. 6. Determinants. 7. Finding inverse matrix. 8. Cramer's rule.
Bibliography:	[Gar07] Garrett, P. Abstract Algebra. Available at: http://www.math.umn.edu/~garrett/m/algebra. [Goo06] Goodman, F., M. Algebra: Abstract and Concrete. Available at: http://homepage.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html. [Sho08] Shoup, V. A Computational Introduction to Number Theory and Algebra. Available at: http://shoup.net/ntb.

Classes in period "Winter Semester 2019/2020" (past)

Time span:	2019-10-01 - 2020-02-23	Selected timetable range: weekly course term Navigate to timetable MO TU W W TH FR CK CK CK
Type of class:	Discussion practice class, 28 hours more information Lecture, 28 hours more information
Coordinators:	Szymon Brzostowski
Group instructors:	Szymon Brzostowski, Kacper Grzelakowski
Students list:	(inaccessible to you)
Examination:	Course - (in Polish) Ocena zgodna z regulaminem studiów Discussion practice class - (in Polish) Ocena zgodna z regulaminem studiów Lecture - (in Polish) Ocena zgodna z regulaminem studiów
(in Polish) Czy IRK BWZ?:	(in Polish) T
Teaching Method:	lecture, talk, discussion, brainstorming
Method and Criteria of Assessment:	The exercise class mark is the average of the marks from two tests. It may be increased in special cases (to students taking active part in the exercises) up to one level up. The lecture mark is the final (written) exam mark. A passing mark for exercise class is a prerequisite for taking the theory exam. The final mark is comprised of the class mark (50%) and the lecture mark (50%).
Course Content:	1. Groups, rings, fields. Operations on complex numbers. 2. Divisibility. Greatest common divisor. Euclidean algorithm and Bezout's theorem. 3. Modular arithmetic, congruences. Linear Diophantine equations. Prime numbers. 4. Matrices, operations in the set of matrices with coefficients in a field. 5. Systems of linear equations. Gauss-Jordan elimination. 6. Determinants. 7. Finding inverse matrix. 8. Cramer's rule.
Bibliography:	[Gar07] Garrett, P. Abstract Algebra. Available at: http://www.math.umn.edu/~garrett/m/algebra. [Goo06] Goodman, F., M. Algebra: Abstract and Concrete. Available at: http://homepage.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html. [Sho08] Shoup, V. A Computational Introduction to Number Theory and Algebra. Available at: http://shoup.net/ntb.

Classes in period "Winter Semester 2018/2019" (past)

Time span:	2018-10-01 - 2019-02-10	Selected timetable range: weekly course term Navigate to timetable MO CK TU W W TH CK FR
Type of class:	Discussion practice class, 28 hours more information Lecture, 28 hours more information
Coordinators:	Szymon Brzostowski
Group instructors:	Szymon Brzostowski, Kacper Grzelakowski
Students list:	(inaccessible to you)
Examination:	Course - (in Polish) Ocena zgodna z regulaminem studiów Discussion practice class - (in Polish) Ocena zgodna z regulaminem studiów Lecture - (in Polish) Ocena zgodna z regulaminem studiów
(in Polish) Czy IRK BWZ?:	(in Polish) T
Teaching Method:	lecture, talk, discussion, brainstorming
Method and Criteria of Assessment:	The exercise class mark is the average of the marks from two tests. It may be increased in special cases (to students taking active part in the exercises) up to one level up. The lecture mark is the final (written) exam mark. A passing mark for exercise class is a prerequisite for taking the theory exam. The final mark is comprised of the class mark (50%) and the lecture mark (50%).
Course Content:	1. Groups, rings, fields. Operations on complex numbers. 2. Divisibility. Greatest common divisor. Euclidean algorithm and Bezout's theorem. 3. Modular arithmetic, congruences. Linear Diophantine equations. Prime numbers. 4. Matrices, operations in the set of matrices with coefficients in a field. 5. Systems of linear equations. Gauss-Jordan elimination. 6. Determinants. 7. Finding inverse matrix. 8. Cramer's rule.
Bibliography:	[Gar07] Garrett, P. Abstract Algebra. Available at: http://www.math.umn.edu/~garrett/m/algebra. [Goo06] Goodman, F., M. Algebra: Abstract and Concrete. Available at: http://homepage.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html. [Sho08] Shoup, V. A Computational Introduction to Number Theory and Algebra. Available at: http://shoup.net/ntb.

Classes in period "Winter Semester 2017/2018" (past)

Time span:	2017-10-01 - 2018-02-09	Selected timetable range: weekly course term Navigate to timetable MO TU W W CK TH FR
Type of class:	Discussion practice class, 28 hours more information Lecture, 28 hours more information
Coordinators:	Szymon Brzostowski
Group instructors:	Szymon Brzostowski
Students list:	(inaccessible to you)
Examination:	Course - (in Polish) Ocena zgodna z regulaminem studiów Discussion practice class - (in Polish) Ocena zgodna z regulaminem studiów Lecture - (in Polish) Ocena zgodna z regulaminem studiów
(in Polish) Czy IRK BWZ?:	(in Polish) T
Teaching Method:	lecture, talk, discussion, brainstorming
Method and Criteria of Assessment:	The exercise class mark is the average of the marks from two tests. It may be increased in special cases (to students taking active part in the exercises) up to one level up. The lecture mark is the final (written) exam mark. A passing mark for exercise class is a prerequisite for taking the theory exam. The final mark is comprised of the class mark (50%) and the lecture mark (50%).
Course Content:	1. Groups, rings, fields. Operations on complex numbers. 2. Divisibility. Greatest common divisor. Euclidean algorithm and Bezout's theorem. 3. Modular arithmetic, congruences. Linear Diophantine equations. Prime numbers. 4. Matrices, operations in the set of matrices with coefficients in a field. 5. Systems of linear equations. Gauss-Jordan elimination. 6. Determinants. 7. Finding inverse matrix. 8. Cramer's rule.
Bibliography:	[Gar07] Garrett, P. Abstract Algebra. Available at: http://www.math.umn.edu/~garrett/m/algebra. [Goo06] Goodman, F., M. Algebra: Abstract and Concrete. Available at: http://homepage.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html. [Sho08] Shoup, V. A Computational Introduction to Number Theory and Algebra. Available at: http://shoup.net/ntb.

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