Number theory

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M2
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

The lecturer selects from the following list of contents:
1. Algebraic numbers: discriminant, algebraic integers, integral basis, norm and trace. Quadratic and cyclotomic fields. Irreducible elements. The problem of unique factorization. Prime elements. Euclidean fields. The Ramanujan-Nagell theorem. Prime factorization.
2. Lattices in Rn . The quotient torus. Minkowski's theorem. Sums of squares. The Dedekind's theorem. Minkowski's constants.
3. The Legendre symbol. Gauss' quadratic reciprocity law. Dirichlet's theorem on primes in arithmetic progression. The Jacobi symbol.
4. Dirichlet's unit theorem. 5. Prime numbers. The sieve of Erathostenes. Testing of factorizability of integers. Pseudoprime numbers. Fermat and Mersenne numbers. Carmichael numbers. The distribution of prime numbers. Regular primes. Heuristic methods. Euler pseudoprimes. Nonlinear diophantine equations. Pythagorean triples. Pell's equation. Kummer's theory of regular primes and Fermat's problem.
6. Lucas sequences. Euler polynomial for irrational numbers. Generating prime numbers. Transcendency of renown numbers. 7. Relative trace and norm. Discriminant and different. Factoring of prime ideals in Galois extensions. The theorem of Kronecker and Weber. The class-field theory.
8. p-adic numbers. Formal power series.

Readings

I. Stewart, D. Tall: Algebraic Number Theory and Fermat’s Last Theorem, AK Peters, Natick, ZDA. 3. izdaja, 2002.
P. Ribenboim: Classical Theory of Algebraic Numbers, Universitext. Springer-Verlag, New York, etc. 2001.
P. Ribenboim: The Little Book of Bigger Primes, Springer-Verlag, New York, etc. 2. izdaja, 2004.
K. H. Rosen: Elementary Number Theory and its Applications, Person, Boston, ZDA. 5. izdaja, 2005.
P. Ribenboim: My Numbers, my Friends, Popular Lectures on Number Theory. Springer-Verlag, New York, etc. 2000.
A. A. Gioia: The Theory of Numbers. An Introduction, Dover Publ. 2001.
S. Alaca, K. S. Williams: Introductory Algebraic Number Theory, Cambridge Univ. Press. 2004.

Objectives and competences

The student learns the basics of the number theory and its applications. The emphasis is on the algebraic theory of numbers.

Intended learning outcomes

Knowledge and understanding:
Knowledge of basic concepts and theorems of the number theory of and their recognition in other areas of mathematics.
Application:
In other areas of mathematics, cryptography and coding theory. Application in computer science and informatics, especially in computer safety
Reflection:
Understanding the theory on the basis of examples and applications.
Transferable skills:
Formulation of problems in appropriate language, solving and analysis of the result on examples, identifying algebraic structures in theory of numbers.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Type (examination, oral, coursework, project):
2 midterm exams instead of written exam, written exam
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Tomaž Košir:
GRUNENFELDER, Luzius, KOŠIR, Tomaž, OMLADIČ, Matjaž, RADJAVI, Heydar. On groups generated by elements of prime order. Geometriae dedicata, ISSN 0046-5755, 1999, let. 75, št. 3, str. 317-332. [COBISS-SI-ID 8849241]
KOŠIR, Tomaž, SETHURAMAN, B. A. Determinantal varieties over truncated polynomial rings. Journal of Pure and Applied Algebra, ISSN 0022-4049. [Print ed.], 2005, vol. 195, no. 1, str. 75-95. [COBISS-SI-ID 13266265]
GRUNENFELDER, Luzius, KOŠIR, Tomaž. Geometric aspect of multiparameter spectral theory. Transactions of the American Mathematical Society, ISSN 0002-9947, 1998, let. 350, št. 6, str. 2525-2546. [COBISS-SI-ID 8449113]