SubjectsSubjects(version: 916)
Course, academic year 2022/2023
   Login via CAS
Algebraic Number Theory and Combinatorics - NDMI066
Title: Algebraická teorie čísel
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2017
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Additional information:
Guarantor: doc. RNDr. Martin Klazar, Dr.
Class: Informatika Mgr. - Diskrétní modely a algoritmy
Classification: Informatics > Discrete Mathematics
Annotation -
Last update: T_KAM (27.04.2005)
If the field of fractions Q is replaced with its finite extension K, for example K=Q(i) or K=Q(2^{1/2}), the ring of integers Z extends in the ring of integers O_K of K. Algebraic number theory investigates arithmetic of O_K, especially unique factorization. These results have important applications in the original ring Z, mainly for solving diophantine equations. In the course we shall present basic notions and results as well as some applications to diophantine equations.
Aim of the course -
Last update: doc. RNDr. Martin Klazar, Dr. (21.09.2016)

The topic is algebraic techniques in combinatorics and number theory.

We will learn many interesting results obtained by them.

Course completion requirements -
Last update: doc. RNDr. Martin Klazar, Dr. (22.09.2020)

Oral exam, in person or in distant mode.

Literature -
Last update: doc. RNDr. Martin Klazar, Dr. (12.10.2017)

Borevich and Shafarevich: Number Theory, Academic Press 1966.

This is one of the references but I draw from many others, such as

the survey article of N. Alon on algebraic methods in combinatorics.

Further literature depends on the current lecture and will be given in the lecture.

Requirements to the exam -
Last update: doc. RNDr. Martin Klazar, Dr. (11.06.2019)

Exam is oral, with written preparation. For concrete requirements see the above homepage of the course.

An example of exam questions:

1. Prove that there are infinitely many primes of the form p = 1 + mn.

2. Prove Wedderburn's theorem on skew fields.

3. Prove Fermat's last theorem ... for polynomials.

4. Prove the theorem of Ko Chao (if q > 3 is a prime number then x^2 - y^q = 1 has no solution in positive integers x, y).

5. Prove the Chevalley-Warning theorem and the corollary on multigraphs.

6. Prove Alon's Combinatorial Nullstellensatz and the corollary on hyperplanes.

Syllabus -
Last update: doc. RNDr. Martin Klazar, Dr. (21.09.2016)

Basic techniques and results from algebra applied in combinatorics and number theory.

For example, in extremal problems (combinatorics) or diophantine equations (number theory).

Charles University | Information system of Charles University |