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Abstract Algebra

Informacje ogólne

Kod przedmiotu: 1100-AA0UMM-Erasm Kod Erasmus / ISCED: (brak danych) / (brak danych)
Nazwa przedmiotu: Abstract Algebra
Jednostka: Wydział Matematyki i Informatyki
Grupy:
Punkty ECTS i inne: 0 LUB 5.00 (w zależności od programu)
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Język prowadzenia: angielski
Efekty kształcenia:

After the course the student:

LO1. formulates definitions, indicates examples and gives outlines of the proofs of the main theorems from the course;

LO2. identifies more specific properties of groups (e.g. simplicity, cyclicity);

LO3. defines basic concepts of the theory of rings and fields and applies their associated properties to solve problems;

LO4. uses the homomorphism theorem to check whether a given algebraic structure forms a group (ring);

LO5. distinguishes specific algebraic structures basing on the understanding of the nature of algebraic properties;

LO6. calculates the GCD of polynomials and solves affine polynomial equations over the ring of polynomials;

LO7. solves problems related to the theory of divisibility in the ring of polynomials basing on the analogy with the theory of divisibility in the ring of integers, and vice versa;

LO8. determines the form of a simple algebraic field extension;

LO9. finds a primitive element of a finite algebraic extension.

Forma studiów:

stacjonarne

Wymagania wstępne:

Knowledge of basic concepts in the scope of the theory of groups, commutative rings and fields.

Skrócony opis:

Celem kursu jest bardziej szczegółowe zapoznanie studentów z głównymi pojęciami i twierdzeniami algebry abstrakcyjnej w zakresie teorii grup, pierścieni przemiennych i ciał.

Zajęcia w cyklu "Semestr zimowy 2017/2018" (w trakcie)

Okres: 2017-10-01 - 2018-02-09
Wybrany podział planu:


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Typ zajęć: Ćwiczenia konwersatoryjne, 28 godzin więcej informacji
Wykład, 28 godzin więcej informacji
Koordynatorzy: Szymon Brzostowski
Prowadzący grup: Szymon Brzostowski, Kacper Grzelakowski
Lista studentów: (nie masz dostępu)
Zaliczenie: Ocena zgodna z regulaminem studiów
Czy IRK BWZ?:

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Metody dydaktyczne:

elements of a lecture, discussion, problem solving

Sposoby i kryteria oceniania:

Evaluation of the problems solved by the student, checking the learning outcomes in practical skills (LO2-LO9).

Evaluation of the learning outcomes in knowledge (LO1) by systematic oral questioning.

The final mark is comprised of the two above marks (50% both).

Treści kształcenia:

1. Groups:

– basic concepts (abelian, cyclic, simple groups; direct sum of groups, group homomorphisms, subgroups, normal subgroups, quotient groups, Lagrange's theorem, the theorems on isomorphisms of groups)

– Cauchy's theorem for abelian groups

– theorem on the element of the maximum order in an abelian group

– theorem on the structure of finitely generated abelian groups

– Sylow theorem


2. Rings (commutative with 1):

– basic concepts (rings, subrings, ideals, quotient rings, ring homomorphisms, the theorem on isomorphism of rings)

– integral domains; theorem on the field of fractions of an integral domain

– prime and maximal ideals; theorem on their characterization

– the concept of a multiplicatively closed set; theorem on the existence of prime ideals


3. The ring of polynomials (especially over a field):

– the concept of GCD and LCM of polynomials; theorem on the division with remainder, Bézout’s identity for polynomials, the existence and uniqueness of GCD, the fundamental theorem of polynomial arithmetic

– prime and irreducible polynomials; the unique factorization theorem for polynomials

– Eisenstein's irreducibility criterion

– Cyclotomic polynomials

– roots of polynomials, the notions of formal derivative and multiplicity of a root of a polynomial

– Gauss' theorem


4. Fields:

– field homomorphisms

– field extensions; the formula for extension degrees in a tower of fields

– algebraic elements, the degree of an algebraic element, the theorem on the form of an extension field produced by adjoining an algebraic element of degree n, Abel's theorem on a primitive element, the concept of a transcendental element

– algebraic extensions, algebraic closure of a field

Literatura:

The basic text:

Garrett, P. - Abstract Algebra. Available at

http://www-users.math.umn.edu/~garrett/m/algebra/notes/Whole.pdf

Supplementary:

Goodman, F., M. - Algebra: Abstract and Concrete. Available at

http://homepage.divms.uiowa.edu/~goodman/algebrabook.dir/book.2.6.pdf

Opisy przedmiotów w USOS i USOSweb są chronione prawem autorskim.
Właścicielem praw autorskich jest Uniwersytet Łódzki.