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.
Lecture 8 (28.05.2019). Polynomials in one variable over a field. Greatest common divisor. Irreducible polynomials. Unique factorization property. Description of ideals. Properties of factor rings.
Lecture 9 (04.06.2019). The characteristic of a field. Extensions of fields. Finite extensions and their degrees. Algebraic and transcendental elements. The minimal polynomial of an algebraic element.
Lecture 10 (11.06.2019). Decomposition of a polynomial into linear factors. Finite fields. Cyclicity of the multiplicative group. Irreducible polynomials over the field $\ZZ_p$. The field with four elements.
The Nth problem sheet contains the Nth homework.
The test has been on Tuesday 11.06.2019, 16:40-19:30. You could use any printed or handwritten notes, a non-programmable calculator.
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