1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. General Algebra

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General Algebra - opis przedmiotu

Cel przedmiotu

In the end of this course the students know and understand the basic theorems concerning groups, rings, fields and lattices theory and they can applicate and use the notions and theorems from the abstract algebra in codes, cryptography and combinatorics.

Wymagania wstępne

Zapisz zmiany

Linear Algebra 1 and 2.

Zakres tematyczny

1. Prime numbers, The Basic Theorem of Arithmetics, congruences of integer numbers, Euler Totient function, Euler theorem. Definitions and properties of operations in the algebraic structures.
2. Groups, abelian groups, cyclic groups, subgroups, permutation groups, torsion and torsion-free groups. Cayley’s theorem and Lagrange’s Thorem for groups. Morphisms of groups, normal subgroups, simple groups, congruences in groups. Quotient groups, Isomorphism theorem for groups. Sylow’s Thorem.
3. Rings, subrings, ideals, congruences in rings, quotient rings. Isomorphism theorem for rings, principal ideals, prime ideals, Maximal ideals. Chinese theorem. Fields, simple fields, finite fields.
4. Polynomial rings in one and many indeterminates, polynomial roots, symmetric polynomials. Bezout’s theorem, Gauss’s theorem, Eisenstein-Shönemann’s criterion. Algebraic elements over a field, minimal polynomial. Extensions of fields. Fields algebraically closed. Hilbert’s zeros Theorem.
5. Lattices, modular and distributive lattices, sublattices, examples. Dedekind-Birkhoff theorem. Boolean algebras.

Metody kształcenia

Lectures: conventional lecture; problem lecture. Exercises: jointly solving exercises related to the subject, proving additional theorems, exercises illustrating the application of theory, problem exercises, discussion.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

The grade for the exercises consists of the results achieved in the tests (80%) and the activity in classes (20%). The exam consists of a written and an oral part To take the oral part, you must obtain at least 30% of the points in the written part. The grade for the course consists of the grade for the exercises (50%) and the grade for the exam (50%). The condition for taking the exam is a positive grade from the exercises. A condition to complete the course, a positive exam grade is required.

Literatura podstawowa

1. David Joyce, Introduction to Modern Algebra, Clark University, 2017 (https://mathcs.clarku.edu/~djoyce/ma225/algebra.pdf)
2. W. J. GILBERT, W. K. NICHOLSON, MODERN ALGEBRA WITH APPLICATIONS, A JOHN WILEY & SONS, INC., PUBLICATION (http://cs.ioc.ee/~margo/aat/Gilbert W.J., Nicholson W.K. Modern algebra with applications (2ed., Wiley, 2004)(ISBN 0471414514)(347s).pdf)

Literatura uzupełniająca

1. S. Burris, H. P. Sankappanavar , A Course in Universal Algebra, (http://orion.math.iastate.edu/cliff/BurrisSanka.pdf)
 

Uwagi

Zmodyfikowane przez dr Joanna Skowronek-Kaziów (ostatnia modyfikacja: 14-01-2024 23:12)