Algebra II *

General data

Course ID:	1000-134AG2*
Erasmus code / ISCED:	11.122 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Algebra II *
Name in Polish:	Algebra II *
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	Elective courses for 1st degree studies in mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	Polish
Type of course:	elective courses
Short description:	Categories and functors, Basic notions of the Theory of Fields including Galois Theory and its applications, Basic notions of Module Theory and multilinear algebra. This lecture covers the topics of Algebra II (1000-134AG2) in a deepened and more general way.
Full description:	1. Categories and functors, isomorphism. Examples: Vect, Top, Gr, Ab, one object categories, sheaves, etc. Product and direct sum in a category, product and direct sum in categories groups and abelian groups. (1-2 lectures) 2. Commutator of elements, commutant of a group, abelianization, Solvable groups, simple groups, solvability of S_n, n < 5, simplicity of A_n, n > 4., semisimple product, exact sequences, splitting, examples (2-3 lectures). 3. Field extensions, the group of automorphism of field extension., algebraic extensions, extensions by a root of a polynomial, splitting field of a polynomial, normal extensions, universal property of normal extensions. The algebraic closure of a field - construction and the uniqueness, Examples. (2-3 lectures). 4. Roots of unity. Existence and uniqueness of fields of p^n elements. 5. Galois Theory of finite extensions in characteristic 0. Irreducible polynomials in characterictic 0 have no multiple roots. Theorem of Abel, Automorphisms of extensions, Galois extensions. Main Theorem of Galois Theory (2-3 lectures). 6. Applications of Galois theory: the Fundamental Theorem of Algebra, solvable extensions, solving equations by radicals (1-2 lectures) 7. Applications of Galois theory: constructability in geometry 8. Modules, torsion elements, direct sum, finitely generated modules, free modules. Homomorphisms of modules, the kermel, factor module, exact sequences of modules, splittings. Classification of finitely generated modules PID. Corollaries :: classification of finitely generated abelian groups, Jordan's Theorem from linear algebra on canonical form of matrices (2-3 lectures). 9. Tensor product of modules, exterior power of a module, exterior algebra. Remark. Topics 4, 7 and 9 are optative. For example, they can be discussed in class only.
Bibliography:	S. Lang, Algebra L. Rowen, Algebra: groups, rings and fields, Wellesley, Massachusetts 1994

This course is not currently offered.

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