UNIWERSYTET ŁÓDZKI - Centralny System Uwierzytelniania
Strona główna

Abstract Algebra

Informacje ogólne

Kod przedmiotu: 1100-AA0UMM-Erasm
Kod Erasmus / ISCED: (brak danych) / (0541) Matematyka Kod ISCED - Międzynarodowa Standardowa Klasyfikacja Kształcenia (International Standard Classification of Education) została opracowana przez UNESCO.
Nazwa przedmiotu: Abstract Algebra
Jednostka: Wydział Matematyki i Informatyki
Grupy:
Punkty ECTS i inne: 0 LUB 5.00 (w zależności od programu) Podstawowe informacje o zasadach przyporządkowania punktów ECTS:
  • roczny wymiar godzinowy nakładu pracy studenta konieczny do osiągnięcia zakładanych efektów uczenia się dla danego etapu studiów wynosi 1500-1800 h, co odpowiada 60 ECTS;
  • tygodniowy wymiar godzinowy nakładu pracy studenta wynosi 45 h;
  • 1 punkt ECTS odpowiada 25-30 godzinom pracy studenta potrzebnej do osiągnięcia zakładanych efektów uczenia się;
  • tygodniowy nakład pracy studenta konieczny do osiągnięcia zakładanych efektów uczenia się pozwala uzyskać 1,5 ECTS;
  • nakład pracy potrzebny do zaliczenia przedmiotu, któremu przypisano 3 ECTS, stanowi 10% semestralnego obciążenia studenta.
zobacz reguły punktacji
Język prowadzenia: angielski
Forma zaliczenia:

egzamin
Poziom studiów:

Studia drugiego stopnia
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.
Efekty uczenia się:

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.

Zajęcia w cyklu "Semestr zimowy 2024/2025" (jeszcze nie rozpoczęty)

Okres: 2024-10-01 - 2025-02-16
Wybrany podział planu:
Przejdź do planu
Typ zajęć:
Ćwiczenia konwersatoryjne, 28 godzin więcej informacji
Wykład, 28 godzin więcej informacji
Koordynatorzy: Szymon Brzostowski
Prowadzący grup: (brak danych)
Lista studentów: (nie masz dostępu)
Zaliczenie: Przedmiot - Ocena zgodna z regulaminem studiów
Ćwiczenia konwersatoryjne - Ocena zgodna z regulaminem studiów
Wykład - Ocena zgodna z regulaminem studiów
Czy IRK BWZ?:

T

Zajęcia w cyklu "Semestr zimowy 2023/2024" (zakończony)

Okres: 2023-10-01 - 2024-02-25
Wybrany podział planu:
Przejdź do planu
Typ zajęć:
Ćwiczenia konwersatoryjne, 28 godzin więcej informacji
Wykład, 28 godzin więcej informacji
Koordynatorzy: Szymon Brzostowski
Prowadzący grup: Szymon Brzostowski
Lista studentów: (nie masz dostępu)
Zaliczenie: Przedmiot - Ocena zgodna z regulaminem studiów
Ćwiczenia konwersatoryjne - Ocena zgodna z regulaminem studiów
Wykład - Ocena zgodna z regulaminem studiów
Czy IRK BWZ?:

T
Metody dydaktyczne:

elements of a lecture, student's presentations, discussion, problem solving, written developments
Sposoby i kryteria oceniania:

Evaluation of the student's solutions to problems checking the learning outcomes in practical skills (LO2-LO9).

The process of evaluating learning outcomes in knowledge (LO1) entails assessing both written developments and oral presentations.

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

Zajęcia w cyklu "Semestr zimowy 2022/2023" (zakończony)

Okres: 2022-10-01 - 2023-02-19
Wybrany podział planu:
Przejdź do planu
Typ zajęć:
Ćwiczenia konwersatoryjne, 28 godzin więcej informacji
Wykład, 28 godzin więcej informacji
Koordynatorzy: Szymon Brzostowski
Prowadzący grup: Szymon Brzostowski
Lista studentów: (nie masz dostępu)
Zaliczenie: Przedmiot - Ocena zgodna z regulaminem studiów
Ćwiczenia konwersatoryjne - Ocena zgodna z regulaminem studiów
Wykład - Ocena zgodna z regulaminem studiów
Czy IRK BWZ?:

T
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

Zajęcia w cyklu "Semestr zimowy 2021/2022" (zakończony)

Okres: 2021-10-01 - 2022-01-23
Wybrany podział planu:
Przejdź do planu
Typ zajęć:
Ćwiczenia konwersatoryjne, 28 godzin więcej informacji
Wykład, 28 godzin więcej informacji
Koordynatorzy: Szymon Brzostowski
Prowadzący grup: Szymon Brzostowski
Lista studentów: (nie masz dostępu)
Zaliczenie: Przedmiot - Ocena zgodna z regulaminem studiów
Ćwiczenia konwersatoryjne - Ocena zgodna z regulaminem studiów
Wykład - Ocena zgodna z regulaminem studiów
Czy IRK BWZ?:

T
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

Zajęcia w cyklu "Semestr zimowy 2020/2021" (zakończony)

Okres: 2020-10-01 - 2021-02-07
Wybrany podział planu:
Przejdź do planu
Typ zajęć:
Ćwiczenia konwersatoryjne, 28 godzin więcej informacji
Wykład, 28 godzin więcej informacji
Koordynatorzy: Szymon Brzostowski
Prowadzący grup: Szymon Brzostowski
Lista studentów: (nie masz dostępu)
Zaliczenie: Przedmiot - Ocena zgodna z regulaminem studiów
Ćwiczenia konwersatoryjne - Ocena zgodna z regulaminem studiów
Wykład - Ocena zgodna z regulaminem studiów
Czy IRK BWZ?:

T
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

Zajęcia w cyklu "Semestr zimowy 2019/2020" (zakończony)

Okres: 2019-10-01 - 2020-02-23
Wybrany podział planu:
Przejdź do planu
Typ zajęć:
Ćwiczenia konwersatoryjne, 28 godzin więcej informacji
Wykład, 28 godzin więcej informacji
Koordynatorzy: Szymon Brzostowski
Prowadzący grup: Szymon Brzostowski
Lista studentów: (nie masz dostępu)
Zaliczenie: Przedmiot - Ocena zgodna z regulaminem studiów
Ćwiczenia konwersatoryjne - Ocena zgodna z regulaminem studiów
Wykład - Ocena zgodna z regulaminem studiów
Czy IRK BWZ?:

T
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

Zajęcia w cyklu "Semestr zimowy 2018/2019" (zakończony)

Okres: 2018-10-01 - 2019-02-10
Wybrany podział planu:
Przejdź do planu
Typ zajęć:
Ćwiczenia konwersatoryjne, 28 godzin więcej informacji
Wykład, 28 godzin więcej informacji
Koordynatorzy: Szymon Brzostowski
Prowadzący grup: Szymon Brzostowski
Lista studentów: (nie masz dostępu)
Zaliczenie: Przedmiot - Ocena zgodna z regulaminem studiów
Ćwiczenia konwersatoryjne - Ocena zgodna z regulaminem studiów
Wykład - Ocena zgodna z regulaminem studiów
Czy IRK BWZ?:

T
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

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

Okres: 2017-10-01 - 2018-02-09
Wybrany podział planu:
Przejdź do planu
Typ zajęć:
Ćwiczenia konwersatoryjne, 28 godzin więcej informacji
Wykład, 28 godzin więcej informacji
Koordynatorzy: Szymon Brzostowski
Prowadzący grup: Szymon Brzostowski
Lista studentów: (nie masz dostępu)
Zaliczenie: Przedmiot - Ocena zgodna z regulaminem studiów
Ćwiczenia konwersatoryjne - Ocena zgodna z regulaminem studiów
Wykład - Ocena zgodna z regulaminem studiów
Czy IRK BWZ?:

T
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.
kontakt deklaracja dostępności USOSweb 7.0.3.0-0