# Algebra DSBA 2018/2019 — различия между версиями

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# Teachers and assistants

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Lecturer Ivan Arzhantsev
Teacher Ivan Arzhantsev Roman Avdeev Nikita Medved

mednik at mccme.ru

Assistant Danila Kutenin kutdanila at yandex.ru Maksim Siplivyj maxsev1999@yandex.ru Ildus Sadrtdinov

# Consultations schedule

Teacher/Assistant Monday Tuesday Wednesday Thursday Friday
1
Ivan Arzhantsev 17:00–18:30, room 603
2
Roman Avdeev 15:40–17:40, room 623 15:40–16:30, 18:10–19:00, room 623
3
Nikita Medved 16:40–18:00, room 623 18:10 (if you write me beforehand)
4
Danila Kutenin 12:00-13:00, room (each time telegram announcement)
5
Maksim Siplivyj 16:40–18:00
6

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 abstracts

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.

# Problem sheets

The Nth problem sheet contains the Nth homework.

# Written test

The test was be on Tuesday 11.06.2019, 16:40-19:30. You could use any printed or handwritten notes, a non-programmable calculator.

# Exam

The exam will be oral.

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