1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. WMIiE - oferta ERASMUS - Program Erasmus
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 to applicate and use the notions and theorems from the abstract algebra in other areas of mathematics.

Wymagania wstępne

Zapisz zmiany

Linear Algebra.

Zakres tematyczny

1. Algebraic structures in Modern Algebra (fields, groups, rings). Isomorphisms, homomorphisms.

2. Prime numbers, divisibility, fundamental theorem of arithmetics, Euclidean algorithm, congruences of integer numbers.

3. Groups, abelian groups, cyclic groups, subgroups, permutation groups, Cayley’s theorem and Lagrange’s thorem for groups. Morphisms of groups, normal subgroups, congruences in groups. Quotient groups, Isomorphism theorem for groups. 

4. Rings, subrings, ideals, congruences in rings, quotient rings. Isomorphism theorem for rings, principal ideals, prime ideals, maximal ideals. Chinese reminder theorem.

5. 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.

6. Fields, subfields, algebraic fields.

7. Posets, lattices, sublattices, distributive lattices, Boolean algebras.

Metody kształcenia

Traditional lectures; Solving appropriate selected exercises in the class.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

Verifying the level of preparation of students and their activities during the classes. The student has to receive the positive grade from two tests with tasks of different difficulty which help to assess whether students have achieved effects of the course in a minimum degree (50% of the final grade). Written exam (50% of the final grade).

Literatura podstawowa

[1]  David Joyce, INTRODUCTION TO MODERN ALGEBRA ,  Clark University, 2017.

[2]  S. Burris, H. P. Sankappanavar , A COURSE IN UNIVERSAL ALGEBRA, ** The Millennium Edition ** , 2000.
 

Literatura uzupełniająca

[1]  W. J. Gilbert and W. K. Nicholson, MODERN ALGEBRA WITH APPLICATIONS, A JOHN WILEY & SONS, 2003.

Uwagi

Zmodyfikowane przez dr Joanna Skowronek-Kaziów (ostatnia modyfikacja: 25-04-2022 13:15)