## MIN401 Mathematics IV

Faculty of Science
Spring 2023
Extent and Intensity
4/2/0. 8 credit(s). Type of Completion: zk (examination).
Taught in person.
Teacher(s)
prof. RNDr. Jan Slovák, DrSc. (lecturer)
Mgr. Martin Doležal (seminar tutor)
Guaranteed by
prof. RNDr. Jan Slovák, DrSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 16:00–17:50 M1,01017, Thu 16:00–17:50 M2,01021
• Timetable of Seminar Groups:
MIN401/01: Thu 10:00–11:50 M3,01023, M. Doležal
Prerequisites
High school mathematics. Elementary knowledge of algebraic and combinatorial tasks (in the extent of MIN101).
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
This is the fourth part of a four semester block of Mathematics. The entire course covers the fundamentals of general algebra and number theory, linear algebra, mathematical analysis, numerical methods, and combinatorics. This semester is concerned with elements of Number theory and its applications and, in the second half, the algebraic concepts and some applications.
Learning outcomes
At the end of this course, students should be able to: understand and use methods of number theory to solve moderately difficult tasks; understand how results of number theory are applied in cryptography; understand basic computational context;
understand algebraic notions and explain general implications and context;
Syllabus
• 1. Number theory (4 weeks) – divisiblity (gcd, extended Euclid algorithm, Bezout); numerics of big numbers (gcd, modular exponential); prime numbers (properties, basic theorems of arithmetics, factorization, prime number testing (Rabin-Miller, Mersenneho prime numbers); congruences (basic properties, small Fermat theorem; Euler theorem; linear congruences; binomial congruences a primitiv roots; discrete logarithm; prime numbers - testing up to AKS, divisors, eliptic curves (introduction); Legendre symbol and the quadratic reciprocity law.
• 2. Number theory applications (2 weeks) – short introduction to asymetric cryptography (RSA, DH, ElGamal, DSA, ECC); basic coding theory (linear and polynomial codes); aplication of fast Fourier transform for quick computations (e.g. Schönhage-Strassen)
• 3. Algebra (7 weeks) – Boolean algebras and lattices, groups, permutations, symetries, modular groups, homomorfisms and factorization, group actions (Burnside lemma); rings and fields (polynomials and their roots, divisibility in integers and in polynomial rings, ideals; finite fields and their basic properties (including applications in computer science; polynomials of more variables (Gröbner basis).
Literature
recommended literature
• J. Slovák, M. Panák a kolektiv, Matematika drsně a svižně, učebnice v přípravě
• RILEY, K.F., M.P. HOBSON and S.J. BENCE. Mathematical Methods for Physics and Engineering. second edition. Cambridge: Cambridge University Press, 2004. 1232 pp. ISBN 0 521 89067 5. info
• GILBERT, William J. and W. Keith NICHOLSON. Modern algebra with applications. 2nd ed. Hoboken, N.J.: Wiley-Interscience, 2004. xvii, 330. ISBN 9780471469889. info
Teaching methods
Four hours of lectures combining theory with problem solving. Seminar groups devoted to solving problems.
Assessment methods
The lectures combining theory with problem solving will be based on material for individual learning, which should precede the lectures. Seminar groups devoted to solving computatinal/practical problems.
Language of instruction
Czech