imported>Dima Trushin: /* Lecture abstracts */

2025-06-24T18:12:16Z

Lecture abstracts

Новая страница

= Teachers and assistants =

{| class="wikitable" style="text-align:center"
|-
! Группа !! 241 !! 242 !! 243 !! 244 !! 245 !! 246
|-
|| Lecturer ||colspan="6"| [https://www.hse.ru/en/org/persons/209508745 Dima Trushin] [https://t.me/DimaTrushin Telegram]
|-
|| Teacher || [https://www.hse.ru/en/org/persons/209508745 Dima Trushin] || [https://www.hse.ru/org/persons/224875083 Andrew Mazhuga] || [https://www.hse.ru/en/org/persons/207901592 Vladislav Balakirev] || [https://www.hse.ru/en/org/persons/392058502 Kirill Shakhmatov] || [https://www.hse.ru/org/persons/780694072/ Aisha Nurieva] || [https://www.hse.ru/en/org/persons/305969353 Alexander Zaytsev]
|-
|| Assistant || [https://t.me/warning172 Roman Bokhyan] || [https://t.me/Avgustine_iw Zakhar Zinkin] || [https://t.me/lunciare Sasha Suvorova] || [https://t.me/tkvlnko Taya Ibragimova] || [https://t.me/meomato Polina Doronicheva] || [https://t.me/rina_mlv Irina Milova]
|}

= Consultations schedule =

{| class="wikitable"
|-
! !! Teacher/Assistant !! How to contact !! When
|-
| <center>1</center> || Dima Trushin || [https://t.me/DimaTrushin telegram] || Wednesday since 17:00 S812
|-
| <center>2</center> || Andrew Mazhuga || [https://www.youtube.com/watch?v=dQw4w9WgXcQ&ab_channel=RickAstley telegram] ||
|-
| <center>3</center> || Vladislav Balakirev || ||
|-
| <center>4</center> || Kirill Shakhmatov || ||
|-
| <center>5</center> || Aisha Nurieva || ||
|-
| <center>6</center> || Alexander Zaytsev || ||
|}

= 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''' (03.04.2025). 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''' (10.04.2025). Classification of cyclic groups. 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''' (17.04.2025). 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''' (24.04.2025). Multiplicative 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''' (15.05.2025). 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''' (22.05.2025). 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''' (29.05.2025). 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''' (05.06.2025). 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''' (09.06.2025). 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''' (19.06.2025). 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. If you send the solution after the deadline your grade will be multiplied by 0.7<sup>t</sup> where t -- is the time passed after the deadline in days (not rounded). So, it is not an issue to send your work 1 or 2 hours after the deadline.

'''Seminar 1''' (03.04.2025). [https://disk.yandex.ru/d/ikcTBq5kJDP1iw Problems]

'''Seminar 2''' (10.04.2025). [https://disk.yandex.ru/d/0da7rDLB2b2O_w Problems]

'''Seminar 3''' (17.04.2025). [https://disk.yandex.ru/i/lsR5vMFIwP9F_Q Problems]

'''Seminar 4''' (24.04.2025). [https://disk.yandex.ru/i/cFA__aXT662WEA Problems]

'''Seminar 5''' (15.05.2025). [https://disk.yandex.ru/i/lzYLUAMfHDBI2Q Problems]

'''Seminar 6''' (22.05.2025). [https://disk.yandex.ru/i/nUvIl-eGV6-clg Problems]

'''Seminar 7''' (29.05.2025). [https://disk.yandex.ru/i/jZFQDVf9apa-1Q Problems]

'''Seminar 8''' (05.06.2025). [https://disk.yandex.ru/i/qPfdD9lx4L4k1A Problems]

= Test =

= Exam =

= Results =

* Homework

{| class="wikitable" style="text-align:center"
|-
! [https://docs.google.com/spreadsheets/d/1Qmhk6OuISvZGx04YgyNQKvk_qqZvR2O8d6sKyADZyNo/edit?gid=640844189#gid=640844189 241] !! [https://docs.google.com/spreadsheets/d/1Qmhk6OuISvZGx04YgyNQKvk_qqZvR2O8d6sKyADZyNo/edit?gid=1612040544#gid=1612040544 242] !! [https://docs.google.com/spreadsheets/d/1Qmhk6OuISvZGx04YgyNQKvk_qqZvR2O8d6sKyADZyNo/edit?gid=1538156963#gid=1538156963 243] !! [https://docs.google.com/spreadsheets/d/1Qmhk6OuISvZGx04YgyNQKvk_qqZvR2O8d6sKyADZyNo/edit?gid=1067620548#gid=1067620548 244] !! [https://docs.google.com/spreadsheets/d/1Qmhk6OuISvZGx04YgyNQKvk_qqZvR2O8d6sKyADZyNo/edit?gid=572193184#gid=572193184 245] !! [https://docs.google.com/spreadsheets/d/1Qmhk6OuISvZGx04YgyNQKvk_qqZvR2O8d6sKyADZyNo/edit?gid=1872255753#gid=1872255753 246]
|}

* Test

{| class="wikitable" style="text-align:center"
|-
! [https://docs.google.com/spreadsheets/d/1SWVSQRerqIrnsDEXWzU2fXIonYIxuZ70FhqOmDjRnWQ/edit?gid=0#gid=0 241] !! [https://docs.google.com/spreadsheets/d/1SWVSQRerqIrnsDEXWzU2fXIonYIxuZ70FhqOmDjRnWQ/edit?gid=450771951#gid=450771951 242] !! [https://docs.google.com/spreadsheets/d/1SWVSQRerqIrnsDEXWzU2fXIonYIxuZ70FhqOmDjRnWQ/edit?gid=1614217882#gid=1614217882 243] !! [https://docs.google.com/spreadsheets/d/1SWVSQRerqIrnsDEXWzU2fXIonYIxuZ70FhqOmDjRnWQ/edit?gid=370973814#gid=370973814 244] !! [https://docs.google.com/spreadsheets/d/1SWVSQRerqIrnsDEXWzU2fXIonYIxuZ70FhqOmDjRnWQ/edit?gid=150412147#gid=150412147 245] !! [https://docs.google.com/spreadsheets/d/1SWVSQRerqIrnsDEXWzU2fXIonYIxuZ70FhqOmDjRnWQ/edit?gid=1833172109#gid=1833172109 246]
|}

* Summary Statement

{| class="wikitable" style="text-align:center"
|-
! [https://docs.google.com/spreadsheets/d/1YPkyaDATcgg-GtZD0uwXft78pmx-Bw64k2x4as4nh1Y/edit?gid=0#gid=0 241] !! [https://docs.google.com/spreadsheets/d/1YPkyaDATcgg-GtZD0uwXft78pmx-Bw64k2x4as4nh1Y/edit?gid=1135622375#gid=1135622375 242] !! [https://docs.google.com/spreadsheets/d/1YPkyaDATcgg-GtZD0uwXft78pmx-Bw64k2x4as4nh1Y/edit?gid=1298301779#gid=1298301779 243] !! [https://docs.google.com/spreadsheets/d/1YPkyaDATcgg-GtZD0uwXft78pmx-Bw64k2x4as4nh1Y/edit?gid=961128608#gid=961128608 244] !! [https://docs.google.com/spreadsheets/d/1YPkyaDATcgg-GtZD0uwXft78pmx-Bw64k2x4as4nh1Y/edit?gid=1640834821#gid=1640834821 245] !! [https://docs.google.com/spreadsheets/d/1YPkyaDATcgg-GtZD0uwXft78pmx-Bw64k2x4as4nh1Y/edit?gid=1692805811#gid=1692805811 246]
|}

= Links =

* Telegram [https://t.me/+h5HG-vxNOxdiNWZi channel] of the course.

* [https://github.com/DimaTrushin/Algebra-DSBA/ Lecture Notes]

DSBA Algebra 2024 2025 - История изменений

imported>Dima Trushin: /* Lecture abstracts */