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Number Theory - NMMB206

Title:	Teorie čísel
Guaranteed by:	Department of Algebra (32-KA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2024
Semester:	summer
E-Credits:	5
Hours per week, examination:	summer s.:2/2, C+Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech
Teaching methods:	full-time
Additional information:	https://www1.karlin.mff.cuni.cz/~kala/web/teaching/24tc

Guarantor:	doc. Mgr. Vítězslav Kala, Ph.D.
Teacher(s):	Bc. Simona Hlavinková doc. Mgr. Vítězslav Kala, Ph.D.
Class:	M Bc. MMIB M Bc. MMIB > Povinně volitelné M Bc. MMIB > 2. ročník M Bc. MMIT M Bc. MMIT > Povinně volitelné M Bc. OM M Bc. OM > Zaměření MSTR M Bc. OM > Povinně volitelné M Bc. OM > 2. ročník
Classification:	Mathematics > Algebra
Incompatibility :	NMIB001
Interchangeability :	NMIB001
Is pre-requisite for:	NMMB349
Is interchangeable with:	NMIB001

Opinion survey results Examination dates SS schedule Noticeboard

Annotation -

In the course we will build up the fundamentals of number theory. We will focus on the existence of primes (estimates of their number, a special case of Dirichlet's theorem on arithmetic progressions), solving Diophantine equations and congruences (Pell equation, quadratic reciprocity), approximations of real numbers using continued fractions, and primality testing by the Rabin-Miller test. Most of these topics non-trivially apply and extend material from the parallel Algebra course.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (21.05.2025)

Course completion requirements - Czech

Zápočet se uděluje za úspěšné vyřešení několika sad domácích úkolů zadaných během semestru (pro detaily viz web). Zápočet není nutnou podmínkou účasti u zkoušky.

Last update: Kala Vítězslav, doc. Mgr., Ph.D. (02.02.2022)

Literature -

lecture notes by Vítězslav Kala: https://www.karlin.mff.cuni.cz/~kala/files/TC25.pdf

Borevič, Šafarevič: Number Theory, Academic Press 1966;

Riesel: Prime numbers and computer methods for factorization, Birkhäuser 1985;

Cohen: A course in computational algebraic number theory, Springer-Verlag 1993.

Last update: Kala Vítězslav, doc. Mgr., Ph.D. (26.02.2025)

Requirements to the exam - Czech

Zkouška bude písemná s několika teoretickými i početními otázkami pokrývajícími látku probranou na přednášce a cvičení.

Detaily viz web kurzu.

Last update: Kala Vítězslav, doc. Mgr., Ph.D. (02.02.2022)

Syllabus -

1. Continued fractions, Pell's equation

2. Characters, quadratic reciprocity, Jacobi symbols

3. Modular arithmetic, Rabin-Miller test, cryptosystem RSA

4. Density of prime numbers, Chebyshev's bound, cyclotomic polynomials

Last update: Stanovský David, doc. RNDr., Ph.D. (22.02.2021)