DSBA Algebra 2022 2023 — различия между версиями
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'''Lecture 5''' (11.05.2023). Rings, commutative rings, fields, subrings. Invertible elements, zero divisors, nilpotent and idempotent elements. Ideals. Description of ideals in Z and Z_n. Homomorphisms and isomorphisms of rings. The Chinese remainder theorem for rings. The kernel and the image of a homomorphism, their properties. | '''Lecture 5''' (11.05.2023). Rings, commutative rings, fields, subrings. Invertible elements, zero divisors, nilpotent and idempotent elements. Ideals. Description of ideals in Z and Z_n. Homomorphisms and isomorphisms of rings. The Chinese remainder theorem for rings. The kernel and the image of a homomorphism, their properties. | ||
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+ | '''Lecture 5''' (18.05.2023). Polynomials in one variable. Euclidean algorithm, greatest common divisor, ideals of F[x]. Irreducible polynomials and unique factorization of polynomials in F[x]. Ring of remainders, the Chinese Remainder Theorem for polynomials. | ||
= Problem sheets = | = Problem sheets = |
Версия 21:00, 20 мая 2023
Содержание
Teachers and assistants
Группа | 221 | 222 | 223 | 224 |
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Lecturer | Dima Trushin Telegram | |||
Teacher | Dima Trushin | Andrew Mazhuga | Nikita Medved | Galina Kaleeva |
Assistant | Misha Gorodov | Dasha Ivanova | Artem Makarenkov |
Consultations schedule
Teacher/Assistant | How to contact | When | |
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Dima Trushin | telegram | Write me and we will schedule a meeting |
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Andrew Mazhuga | telegram | |
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Nikita Medved | ||
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Galina Kaleeva | Tuesday, 18:00, zoom. Please notify me beforehand | |
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Misha Gorodov | telegram | |
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Dasha Ivanova | telegram | |
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Artem Makarenkov | telegram | |
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Alena Chislova | telegram |
Grading system
The final grade is computed as follows
F = 0,3 * H + 0,3 T + 0,4 E
where H is the grade for the home assignments, T is the written test grade, and E is the final exam grade.
Only the final grade is rounded in the final formula according to the standard rule.
Lecture abstracts
Lecture 1 (06.04.2023). Binary operations. Associativity, neutral element, inverse element, commutativity. Definition of a group. Additive and multiplicative notations. Subgroups and cyclic subgroups. The order of an element of a group. Classification of cyclic groups.
Lecture 2 (13.04.2023). The subgroups of the group of integers. The subgroups of the group Z_n. Left and right cosets, examples. Normal subgroups. The Lagrange theorem and its corollaries.
Lecture 3 (20.04.2023). Homomorphisms and Isomorphisms of groups. Image and kernel of a homomorphism. Normal subgroups. Direct product of groups. Finite Abelian Groups. The Chinese Remainder Theorem. Structure of a finite abelian group.
Lecture 4 (27.04.2023). Second version of the Chinese Remainder Theorem. Structure of Z_{p^n}^*. Cryptography. Exponentiation by squaring (fast raising to a power algorithm). The discrete logarithm problem. Diffie-Hellman key exchange. RSA.
Lecture 5 (11.05.2023). Rings, commutative rings, fields, subrings. Invertible elements, zero divisors, nilpotent and idempotent elements. Ideals. Description of ideals in Z and Z_n. Homomorphisms and isomorphisms of rings. The Chinese remainder theorem for rings. The kernel and the image of a homomorphism, their properties.
Lecture 5 (18.05.2023). Polynomials in one variable. Euclidean algorithm, greatest common divisor, ideals of F[x]. Irreducible polynomials and unique factorization of polynomials in F[x]. Ring of remainders, the Chinese Remainder Theorem for polynomials.
Problem sheets
The solutions should be sent to your teaching assistant before the beginning of the next seminar. The deadline is strict. We do not evaluate the homework sent after the deadline.
Seminar 1 (06.04.2023). Problems
Seminar 2 (13.04.2023). Problems
Seminar 3 (20.04.2023). Problems
Seminar 4 (27.04.2023). Problems
Seminar 5 (11.05.2023). Problems
Test
Exam
Results
- Homework
221 | 222 | 223 | 224 |
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Links
- Telegram chat of the course.
DSBA 2022/2023 |
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First year |