Algebra DSBA 2020/2021 — различия между версиями
Mednik (обсуждение | вклад) |
(→Exam) |
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(не показано 20 промежуточных версии 3 участников) | |||
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| <center>3</center> || Nikita Medved || || || write me in telegram https://t.me/medvednikita | | <center>3</center> || Nikita Medved || || || write me in telegram https://t.me/medvednikita | ||
− | + | and we will schedule a meeting | |
+ | || || | ||
|- | |- | ||
| <center>4</center> || Galina Kaleeva || || || 16.20-17.40. | | <center>4</center> || Galina Kaleeva || || || 16.20-17.40. | ||
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'''Lecture 1''' (08.04.2021). 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. | '''Lecture 1''' (08.04.2021). 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. | ||
− | '''Lecture 2''' (15.04. | + | '''Lecture 2''' (15.04.2021). Subgroups of the group of integers. Left and right cosets, examples. Normal subgroups. The Lagrange theorem. |
− | '''Lecture 3''' (22.04. | + | '''Lecture 3''' (22.04.2021). Five corollaries of the Lagrange theorem. 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''' (29.04.2021). 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. | ||
+ | |||
+ | '''Lecture 5''' (13.05.2021). 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 6''' (20.05.2021). 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. | ||
+ | |||
+ | '''Lecture 7''' (27.05.2021). Characteristic of a field. Field extensions, an extension by a root. Finite fields: number of elements in a finite field, multiplicative group of a finite field is cyclic, classification of finite fields (without proof). How to produce finite fields. Galois random generator. Stream cipher. | ||
+ | |||
+ | '''Lecture 8''' (03.06.2021). Polynomials in several variables. Lexicographical orders, stabilization of strictly descending chains of monomials. An elementary reduction, a reduction with respect to a set of polynomials, remainders, Groebner basis. Stabilization of reduction. | ||
+ | |||
+ | '''Lecture 9''' (10.06.2021). S-polynomials and the Buchberger criterion. Ideals in a polynomial ring, the Buchberger algorithm to produce a Groebner basis of an ideal. A ring of remainders. Membership problem and variable elimination. | ||
+ | |||
+ | '''Lecture 10''' (17.06.2021). The Diamond Lemma. A proof of the Buchberger criterion. | ||
= Problem sheets = | = Problem sheets = | ||
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'''Seminar 3''' (22.04.2021). [https://disk.yandex.ru/i/druoZuJJTfUsLg Problems] | '''Seminar 3''' (22.04.2021). [https://disk.yandex.ru/i/druoZuJJTfUsLg Problems] | ||
+ | |||
+ | '''Seminar 4''' (29.04.2021). [https://disk.yandex.ru/i/ObhKt0v6Kj-d4A Problems] | ||
+ | |||
+ | '''Seminar 5''' (13.05.2021). [https://disk.yandex.ru/i/2cHws9wRUMkNoA Problems] | ||
+ | |||
+ | '''Seminar 6''' (20.05.2021). [https://disk.yandex.ru/i/P07PYPjuxnbX9A Problems] | ||
+ | |||
+ | Since Homework 6 is late, there is a special deadline for Group 201. The new deadline is Sunday 23:00 May 30. Deadlines for the other groups remain the same. | ||
+ | |||
+ | '''Seminar 7''' (27.05.2021). [https://disk.yandex.ru/i/v-gwT7_dvGvx2w Problems] | ||
+ | |||
+ | '''Seminar 8''' (03.06.2021). [https://disk.yandex.ru/i/e3OvTAA6MA0dOA Problems] Deadline for group 204: Friday, 18.06.2021, 14.40. | ||
+ | |||
+ | '''Seminar 9''' (10.06.2021). [https://disk.yandex.ru/i/AErL_P-U1FinzA Problems] | ||
+ | |||
+ | = Test = | ||
+ | |||
+ | Demo variant of the [https://disk.yandex.ru/i/GPUiapfVUdzc6g test] | ||
= Exam = | = Exam = | ||
+ | |||
+ | * The [https://disk.yandex.ru/i/smwqxW5UXQxGBA link] to the rules and information regarding the exam. | ||
= Results = | = Results = | ||
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|- | |- | ||
! [https://docs.google.com/spreadsheets/d/11y6e-yqHgs12Qka_vdvR9NOUZiX9PmF1WrnP3HX6Uro/edit#gid=0 201] !! [https://docs.google.com/spreadsheets/d/11y6e-yqHgs12Qka_vdvR9NOUZiX9PmF1WrnP3HX6Uro/edit#gid=580483061 202] !! [https://docs.google.com/spreadsheets/d/11y6e-yqHgs12Qka_vdvR9NOUZiX9PmF1WrnP3HX6Uro/edit#gid=1410164784 203] !! [https://docs.google.com/spreadsheets/d/11y6e-yqHgs12Qka_vdvR9NOUZiX9PmF1WrnP3HX6Uro/edit#gid=133833186 204] | ! [https://docs.google.com/spreadsheets/d/11y6e-yqHgs12Qka_vdvR9NOUZiX9PmF1WrnP3HX6Uro/edit#gid=0 201] !! [https://docs.google.com/spreadsheets/d/11y6e-yqHgs12Qka_vdvR9NOUZiX9PmF1WrnP3HX6Uro/edit#gid=580483061 202] !! [https://docs.google.com/spreadsheets/d/11y6e-yqHgs12Qka_vdvR9NOUZiX9PmF1WrnP3HX6Uro/edit#gid=1410164784 203] !! [https://docs.google.com/spreadsheets/d/11y6e-yqHgs12Qka_vdvR9NOUZiX9PmF1WrnP3HX6Uro/edit#gid=133833186 204] | ||
+ | |} | ||
+ | |||
+ | * Summary Statement | ||
+ | |||
+ | {| class="wikitable" style="text-align:center" | ||
+ | |- | ||
+ | ! [https://docs.google.com/spreadsheets/d/1qIcyiaujcd8DCYoClJQ70dqbZHIONdM4j1ERpTJgL84/edit#gid=0 201] !! [https://docs.google.com/spreadsheets/d/1qIcyiaujcd8DCYoClJQ70dqbZHIONdM4j1ERpTJgL84/edit#gid=312933063 202] !! [https://docs.google.com/spreadsheets/d/1qIcyiaujcd8DCYoClJQ70dqbZHIONdM4j1ERpTJgL84/edit#gid=1854307677 203] !! [https://docs.google.com/spreadsheets/d/1qIcyiaujcd8DCYoClJQ70dqbZHIONdM4j1ERpTJgL84/edit#gid=1284918112 204] | ||
|} | |} | ||
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* [https://disk.yandex.ru/d/Bk0bacA5HuxA8w?w=1 '''Seminars 201'''] | * [https://disk.yandex.ru/d/Bk0bacA5HuxA8w?w=1 '''Seminars 201'''] | ||
+ | |||
+ | * Two seminars 203 online: [https://youtu.be/lhR08fZbDy0 '''8'''], [https://youtu.be/OTR6Y2hLlCY '''9'''] and corresponding whiteboards: [https://miro.com/app/board/o9J_l__r74c=/ '''8'''], [https://miro.com/app/board/o9J_l_YHP9Y=/ '''9'''] |
Текущая версия на 12:09, 14 июня 2022
Содержание
Schedule
- Lecture Thursday 16:20–17:40
- Seminar 201 Thursday 18:10–19:30
Teachers and assistants
Группа | 201 | 202 | 203 | 204 |
---|---|---|---|---|
Lecturer | Dima Trushin Telegram | |||
Teacher | Dima Trushin | Andrew Mazhuga | Nikita Medved | Galina Kaleeva |
Assistant | Masha Marchenko | Daniil Kopytov | Ваня Пешехонов | Dasha Ivanova |
Consultations schedule
Teacher/Assistant | Monday | Tuesday | Wednesday | Thursday | Friday | |
---|---|---|---|---|---|---|
|
Dima Trushin | zoom since 17:00 | ||||
|
Andrew Mazhuga | |||||
|
Nikita Medved | write me in telegram https://t.me/medvednikita
and we will schedule a meeting |
||||
|
Galina Kaleeva | 16.20-17.40.
Offline: S 913 Online: Zoom Passcode: algebra In case I'm offline text me via telegram |
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|
Masha Marchenko | |||||
|
Daniil Kopytov | |||||
|
Vanya Peshekhonov | zoom c 18:20 | ||||
|
Dasha Ivanova |
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 (08.04.2021). 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.
Lecture 2 (15.04.2021). Subgroups of the group of integers. Left and right cosets, examples. Normal subgroups. The Lagrange theorem.
Lecture 3 (22.04.2021). Five corollaries of the Lagrange theorem. 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 (29.04.2021). 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.
Lecture 5 (13.05.2021). 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 6 (20.05.2021). 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.
Lecture 7 (27.05.2021). Characteristic of a field. Field extensions, an extension by a root. Finite fields: number of elements in a finite field, multiplicative group of a finite field is cyclic, classification of finite fields (without proof). How to produce finite fields. Galois random generator. Stream cipher.
Lecture 8 (03.06.2021). Polynomials in several variables. Lexicographical orders, stabilization of strictly descending chains of monomials. An elementary reduction, a reduction with respect to a set of polynomials, remainders, Groebner basis. Stabilization of reduction.
Lecture 9 (10.06.2021). S-polynomials and the Buchberger criterion. Ideals in a polynomial ring, the Buchberger algorithm to produce a Groebner basis of an ideal. A ring of remainders. Membership problem and variable elimination.
Lecture 10 (17.06.2021). The Diamond Lemma. A proof of the Buchberger criterion.
Problem sheets
The solutions should be sent to your teaching assistant before the beginning of the next seminar.
Seminar 1 (08.04.2021). Problems
Seminar 2 (15.04.2021). Problems
Seminar 3 (22.04.2021). Problems
Seminar 4 (29.04.2021). Problems
Seminar 5 (13.05.2021). Problems
Seminar 6 (20.05.2021). Problems
Since Homework 6 is late, there is a special deadline for Group 201. The new deadline is Sunday 23:00 May 30. Deadlines for the other groups remain the same.
Seminar 7 (27.05.2021). Problems
Seminar 8 (03.06.2021). Problems Deadline for group 204: Friday, 18.06.2021, 14.40.
Seminar 9 (10.06.2021). Problems
Test
Demo variant of the test
Exam
- The link to the rules and information regarding the exam.
Results
- Homework
201 | 202 | 203 | 204 |
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- Summary Statement
201 | 202 | 203 | 204 |
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