Abstract Algebra
Informacje ogólne
Kod przedmiotu: | 1100-AA0UMM-Erasm |
Kod Erasmus / ISCED: |
(brak danych)
/
(0541) Matematyka
|
Nazwa przedmiotu: | Abstract Algebra |
Jednostka: | Wydział Matematyki i Informatyki |
Grupy: | |
Punkty ECTS i inne: |
0 LUB
5.00
(w zależności od programu)
|
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 |
Przejdź do planu
PN WT ŚR CZ PT |
Typ zajęć: |
Ćwiczenia konwersatoryjne, 28 godzin
Wykład, 28 godzin
|
|
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 |
Przejdź do planu
PN WT W
ŚR CZ CK
PT |
Typ zajęć: |
Ćwiczenia konwersatoryjne, 28 godzin
Wykład, 28 godzin
|
|
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 |
Przejdź do planu
PN W
WT ŚR CK
CZ W
PT |
Typ zajęć: |
Ćwiczenia konwersatoryjne, 28 godzin
Wykład, 28 godzin
|
|
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 |
Przejdź do planu
PN WT CK
ŚR CZ PT W
|
Typ zajęć: |
Ćwiczenia konwersatoryjne, 28 godzin
Wykład, 28 godzin
|
|
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 |
Przejdź do planu
PN WT ŚR CZ PT |
Typ zajęć: |
Ćwiczenia konwersatoryjne, 28 godzin
Wykład, 28 godzin
|
|
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 |
Przejdź do planu
PN WT ŚR CZ PT CK
W
|
Typ zajęć: |
Ćwiczenia konwersatoryjne, 28 godzin
Wykład, 28 godzin
|
|
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 |
Przejdź do planu
PN WT W
CK
ŚR CZ PT |
Typ zajęć: |
Ćwiczenia konwersatoryjne, 28 godzin
Wykład, 28 godzin
|
|
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 |
Przejdź do planu
PN WT ŚR CZ PT |
Typ zajęć: |
Ćwiczenia konwersatoryjne, 28 godzin
Wykład, 28 godzin
|
|
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 |
Właścicielem praw autorskich jest UNIWERSYTET ŁÓDZKI.