Algebra II - NALG027
| Title: |
Algebra II |
| Guaranteed by: |
Department of Algebra (32-KA) |
| Faculty: |
Faculty of Mathematics and Physics |
| Actual: |
from 2018 |
| Semester: |
summer |
| E-Credits: |
3 |
| Hours per week, examination: |
summer s.:2/0, Ex [HT] |
| Capacity: |
unlimited |
| Min. number of students: |
unlimited |
| 4EU+: |
no |
| Virtual mobility / capacity: |
no |
| State of the course: |
cancelled |
| Language: |
Czech |
| Teaching methods: |
full-time |
| Teaching methods: |
full-time |
|
| Guarantor: |
prof. RNDr. Jan Trlifaj, CSc., DSc. |
| Classification: |
Mathematics > Algebra |
| Pre-requisite : |
{Linear Algebra and Geometry} |
| Co-requisite : |
NALG026 |
| Interchangeability : |
NMAG202, NMAI063 |
| Is co-requisite for: |
NALG008, NALG009, NALG006 |
| Is incompatible with: |
NMAG202, NMAX063, NUMP007, NMAI063, NMUE004, NUMZ004, NUMP020, NUMP019, NMUE033 |
| Is pre-requisite for: |
NALG019, NALG013, NALG012, NALG007, NALG071, NALG072, NALG067, NALG070, NALG078 |
| Is interchangeable with: |
NMUE033, NUMP007, NUMP019, NMUE004, NMAG202, NUMP020, NUMZ004 |
|
| Annotation -
| |
|
Last update: G_M (02.06.2011)
Basic concepts and results of commutative algebra. Introduction to Boolean algebras.
Last update: G_M (02.06.2011)
Základní pojmy a věty komutativní algebry. Úvod do Booleových algeber.
|
| Literature -
| |
|
Last update: T_KA (19.05.2010)
S.Lang, Algebra, Revised 3rd ed., GTM 211, Springer, New York, 2002.
N. Lauritzen, Concrete Abstract Algebra, Cambridge Univ. Press, Cambridge 2003.
C. Menini and F. van Oystaeyen: ``Abstract Algebra'', M. Dekker, New York 2004.
L.Procházka a kol., Algebra, Academia, Praha, 1990 (in Czech).
J.Trlifaj, Algebra II, http://www.karlin.mff.cuni.cz/~trlifaj/NALG027.pdf (in Czech).
Last update: T_KA (19.05.2010)
S.Lang, Algebra, Revised 3rd ed., GTM 211, Springer, New York, 2002.
N. Lauritzen, Concrete Abstract Algebra, Cambridge Univ. Press, Cambridge 2003.
C. Menini a F. van Oystaeyen: ``Abstract Algebra'', M. Dekker, New York 2004.
L.Procházka a kol., Algebra, Academia, Praha, 1990.
J.Trlifaj, Algebra II, http://www.karlin.mff.cuni.cz/~trlifaj/NALG027.pdf
|
| Syllabus -
| |
|
Last update: T_KA (20.05.2010)
1. Polynomial rings.
1.1 Divisibility for integral domains. Hilbert Basis Theorem .
1.2 UFD's and Euclidean domains, The Euclid algorithm.
1.3 Derivation and multiplicity of roots, perfect fields.
1.4 Symmetric polynomials, the Main Theorem, and its applications.
2. Fields.
2.1 Field extensions of finite degree.
2.2 Splitting fields, their existence and uniqueness, algebraic closure.
2.3 The structure of finite fields.
3. Lattices and Boolean algebras.
3.1 Complete and modular lattices.
3.2 Boolean algebras, structure of finite Boolean algebras.
Suplementary topic: Introduction to universal algebra. Terms and free algebras.
Last update: T_KA (19.05.2010)
1. Okruhy polynomů.
1.1 Dělitelnost v oborech integrity, Hilbertova věta o bázi.
1.2 Gaussovy a eukleidovské obory, Eukleidův algoritmus.
1.3 Derivace a násobnost kořenů, perfektní tělesa.
1.4 Symetrické polynomy, hlavní věta o nich a její aplikace.
2. Komutativní tělesa.
2.1 Rozšíření konečného stupně.
2.2 Kořenová a rozkladová nadtělesa, jejich existence a jednoznačnost, algebraický uzávěr.
2.3 Struktura konečných těles.
3. Svazy a Booleovy algebry.
3.1 Úplné a modulární svazy.
3.2 Booleovy algebry, struktura konečných Booleových algeber.
Rozšiřující téma: Úvod do univerzální algebry. Termy a volné algebry.
|
|
