Algebra DSBA 2018/2019
Teachers and assistants
|Teacher||Ivan Arzhantsev||Roman Avdeev|| Nikita Medved
mednik at mccme.ru
|Assistant||Danila Kutenin kutdanila at yandex.ru||Maksim Siplivyj firstname.lastname@example.org|| Ildus Sadrtdinov
||Ivan Arzhantsev||17:00–18:30, room 603|
||Roman Avdeev||15:40–17:40, room 623||15:40–16:30, 18:10–19:00, room 623|
||Nikita Medved||16:40–18:00, room 623||18:10 (if you write me beforehand)|
||Danila Kutenin||12:00-13:00, room (each time telegram announcement)|
The cumulative grade is computed as follows:
C = 0,6 * H + 0,4 * T,
where H is the grade for the home assignments and T is the written test grade.
The final course grade is given by
F = 0,5 * C + 0,5 * E
where E is the final exam grade.
Grades in all formulas are rounded according to the standard rule.
Lecture 1 (2.04.2019). Semigroups and groups: definitions and examples. Permutation groups and matrix groups. Subgroups. The order of an element and cyclic subgroups.
Lecture 2 (9.04.2019). Lagrange's theorem and its corollaries. Normal subgroups. Homomorphisms and isomorphisms. A classification of cyclic groups. Factor groups and the Homomorphism theorem.
Lecture 3 (16.04.2019). The homomorphism theorem. The center and direct products of groups. Theorem on factorization of direct products and factorization of finite cyclic groups.
Lecture 4 (23.04.2019). Free abelian groups and their subgroups. Stacked bases. An algorithm for transforming an integer matrix to a diagonal form. Classification of finite abelian groups. The exponent of a finite abelian group.
Lecture 5 (30.04.2019). Actions of a group on a set. Orbits and stabilizers. Transitive actions and free actions. Three actions of a group on itself. Conjugacy classes. Cayley's Theorem.
Lecture 6 (14.05.2019). Rings and fields. Zero divisors, invertible elements, nilpotents and idempotents. Ideals. Principal ideals. Factor rings and the Homomorphism Theorem.
Lecture 7 (21.05.2019). Polynomials in several variables. Symmetric polynomials. The lexicographic order. Elementary symmetric polynomials. The main theorem on symmetric polynomials. Vieta's formulas. The discriminant.
The Nth problem sheet contains the Nth homework.
The exam will be oral.
- Э.Б.Винберг. Курс алгебры. М.: МЦНМО, 2014 (English transl.: Ernest Vinberg. A Course in Algebra. Graduate Studies in Math. 56, Amer. Math. Soc., 2003)
- Сборник задач по алгебре под редакцией А.И.Кострикина. Новое издание. М.: МЦНМО, 2015 (English transl.: Exercises in Algebra. Edited by A. Kostrikin, CRC Press, 1996)
Serge Lang. Algebra. Revised Third Edition. Graduate Texts in Math. 211, Springer, 2002