DSBA Algebra 2022 2023 — различия между версиями

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(не показано 30 промежуточных версии 4 участников)
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|| Assistant || [https://t.me/m_gorodov Misha Gorodov] || [https://t.me/ivanovskayaaaaa Dasha Ivanova] || [https://t.me/Artem_Makarenkov Artem Makarenkov] ||  
 
|| Assistant || [https://t.me/m_gorodov Misha Gorodov] || [https://t.me/ivanovskayaaaaa Dasha Ivanova] || [https://t.me/Artem_Makarenkov Artem Makarenkov] ||  
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[https://t.me/alenka_chislova Alena Chislova]
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[https://t.me/Alyona_Chislova Alena Chislova]
 
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| <center>3</center> || Nikita Medved ||  ||  
 
| <center>3</center> || Nikita Medved ||  ||  
 
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| <center>4</center> || Galina Kaleeva ||  || Tuesday, 18:00, [https://yandex.zoom.us/j/93497715159?pwd=ZmRmbHNvbEVWUmwwaVZNRnl3TmNIdz09 zoom] please notify me beforehand
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| <center>4</center> || Galina Kaleeva ||  || Tuesday, 18:00, [https://yandex.zoom.us/j/93497715159?pwd=ZmRmbHNvbEVWUmwwaVZNRnl3TmNIdz09 zoom]. Please notify me beforehand
 
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| <center>5</center> || Misha Gorodov  || [https://t.me/m_gorodov telegram] ||
 
| <center>5</center> || Misha Gorodov  || [https://t.me/m_gorodov telegram] ||
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| <center>7</center> || Artem Makarenkov || [https://t.me/Artem_Makarenkov telegram]  ||
 
| <center>7</center> || Artem Makarenkov || [https://t.me/Artem_Makarenkov telegram]  ||
 
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| <center>8</center> || Alena Chislova || [https://t.me/alenka_chislova telegram] ||
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| <center>8</center> || Alena Chislova || [https://t.me/Alyona_Chislova telegram] || Write me and we will schedule a meeting
 
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'''Lecture&nbsp;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&nbsp;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.
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'''Lecture&nbsp;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.
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'''Lecture&nbsp;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.
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'''Lecture&nbsp;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&nbsp;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&nbsp;6''' (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.
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'''Lecture&nbsp;7''' (25.05.2023). 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.
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'''Lecture&nbsp;8''' (01.06.2023). 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&nbsp;9''' (08.06.2023). 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.
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'''Lecture&nbsp;10''' (15.06.2023). The Diamond Lemma. A proof of the Buchberger criterion. The Dickson Lemma and termination of the Buchberger algorithm.
  
 
= Problem sheets =
 
= Problem sheets =
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'''Seminar&nbsp;1''' (06.04.2023). [https://disk.yandex.ru/i/XgLEAuGENj60oA Problems]
 
'''Seminar&nbsp;1''' (06.04.2023). [https://disk.yandex.ru/i/XgLEAuGENj60oA Problems]
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'''Seminar&nbsp;2''' (13.04.2023). [https://disk.yandex.ru/i/BV7aSHn2j4x8cA Problems]
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'''Seminar&nbsp;3''' (20.04.2023). [https://disk.yandex.ru/i/du_S-hrpUq1CHw Problems]
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'''Seminar&nbsp;4''' (27.04.2023). [https://disk.yandex.ru/i/X6t_ne0ucnSkFg Problems]
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'''Seminar&nbsp;5''' (11.05.2023). [https://disk.yandex.ru/i/iJmYYdMEuEJu3w Problems]
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'''Seminar&nbsp;6''' (18.05.2023). [https://disk.yandex.ru/i/8Fe__0Z53zwXJg Problems]
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'''Seminar&nbsp;7''' (25.05.2023). [https://disk.yandex.ru/i/SDX31aIDcrx36w Problems]
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'''Seminar&nbsp;8''' (01.06.2023). [https://disk.yandex.ru/i/p5XGbnJX4minNg Problems]
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'''Seminar&nbsp;9''' (08.06.2023). [https://disk.yandex.ru/i/VJ-yW3lKrg-qoA Problems]
  
 
= Test =
 
= Test =
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The test will take place on Monday 19 of June, since 10:00 in online format. The following file contains all the information.
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* [https://disk.yandex.ru/i/8MPBRsygXuXaPw Rules]
  
 
= Exam =
 
= Exam =
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The exam will take place on June 24, Saturday.
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* [https://disk.yandex.ru/i/J898MXSssf8NlQ List] of definitions and statements.
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* [https://disk.yandex.ru/i/xS-YzNLp4b0SEQ List] of statements to prove.
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* The [https://disk.yandex.ru/i/D1D_z4btcSpMVQ rules] for the exam.
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The [https://docs.google.com/spreadsheets/d/1Oq23V0q_8HYEFL12sD9g_-qqZbIAzLoF6de9VqXDZIc/edit#gid=0 schedule] for the exam. You must come at the time in the schedule.
  
 
= Results =
 
= Results =
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! [https://docs.google.com/spreadsheets/d/1HE6IkwUldYJ3Dt1ugXZzstFTpOpoO5XOh_Ln1O5tXRA/edit#gid=0 221] !! [https://docs.google.com/spreadsheets/d/1HE6IkwUldYJ3Dt1ugXZzstFTpOpoO5XOh_Ln1O5tXRA/edit#gid=1485348044 222] !! [https://docs.google.com/spreadsheets/d/1HE6IkwUldYJ3Dt1ugXZzstFTpOpoO5XOh_Ln1O5tXRA/edit#gid=826068042 223] !! [https://docs.google.com/spreadsheets/d/1HE6IkwUldYJ3Dt1ugXZzstFTpOpoO5XOh_Ln1O5tXRA/edit#gid=1061162416 224]
 
! [https://docs.google.com/spreadsheets/d/1HE6IkwUldYJ3Dt1ugXZzstFTpOpoO5XOh_Ln1O5tXRA/edit#gid=0 221] !! [https://docs.google.com/spreadsheets/d/1HE6IkwUldYJ3Dt1ugXZzstFTpOpoO5XOh_Ln1O5tXRA/edit#gid=1485348044 222] !! [https://docs.google.com/spreadsheets/d/1HE6IkwUldYJ3Dt1ugXZzstFTpOpoO5XOh_Ln1O5tXRA/edit#gid=826068042 223] !! [https://docs.google.com/spreadsheets/d/1HE6IkwUldYJ3Dt1ugXZzstFTpOpoO5XOh_Ln1O5tXRA/edit#gid=1061162416 224]
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* Test
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{| class="wikitable" style="text-align:center"
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! [https://docs.google.com/spreadsheets/d/15gFIaDLl6tr3I67ftEhc9cuNlUPUR2PmoleWE67Q0vo/edit#gid=0 221] !! [https://docs.google.com/spreadsheets/d/15gFIaDLl6tr3I67ftEhc9cuNlUPUR2PmoleWE67Q0vo/edit#gid=450771951 222] !! [https://docs.google.com/spreadsheets/d/15gFIaDLl6tr3I67ftEhc9cuNlUPUR2PmoleWE67Q0vo/edit#gid=1614217882 223] !! [https://docs.google.com/spreadsheets/d/15gFIaDLl6tr3I67ftEhc9cuNlUPUR2PmoleWE67Q0vo/edit#gid=370973814 224]
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|}
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* Summary Statement
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{| class="wikitable" style="text-align:center"
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|-
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! [https://docs.google.com/spreadsheets/d/1Cj85xg4jqgD_G3l2uBWgcPC15lHsQrNL8uYl_KGv5F4/edit#gid=0 221] !! [https://docs.google.com/spreadsheets/d/1Cj85xg4jqgD_G3l2uBWgcPC15lHsQrNL8uYl_KGv5F4/edit#gid=1530981910 222] !! [https://docs.google.com/spreadsheets/d/1Cj85xg4jqgD_G3l2uBWgcPC15lHsQrNL8uYl_KGv5F4/edit#gid=1340480276 223] !! [https://docs.google.com/spreadsheets/d/1Cj85xg4jqgD_G3l2uBWgcPC15lHsQrNL8uYl_KGv5F4/edit#gid=1432577033 224]
 
|}
 
|}
  
 
= Links =
 
= Links =
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* Telegram [https://t.me/+EGHoT8x-nDozMWYy chat] of the course.
  
 
* [https://github.com/DimaTrushin/Algebra-DSBA Lecture Notes]
 
* [https://github.com/DimaTrushin/Algebra-DSBA Lecture Notes]
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= Navigation =
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<div style="border:1px; border-style:solid; border-color:#a2a9b1; padding:3px">
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<table style="border-spacing:0; background:transparent; color:inherit; width:100%">
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<tr>
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  <th scope="colgroup" style="background-color:#cfe3ff" colspan="2"><div id="DSBA22" style="font-size:114%; margin:0 5em">DSBA 2022/2023</div></th>
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</tr>
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<tr >
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  <th scope="row" style="border-top-width:2px; border-top-color:#fdfdfd; border-top-style:solid; width:10%; background-color:#dcebff">First year</th>
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  <td style="text-align:left; border-left-width:2px; border-left-style:solid; border-color:#fdfdfd; width:90%; padding:0px">
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  <div style="padding:0em 0.25em">
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  <ul><li style="display:inline-block; padding:2px">[[LAaG_DSBA_2022/2023|Linear Algebra and Geometry]] &bull; </li><li style="display:inline-block; padding:2px">[[Discrete_Mathematics_DSBA_2022/2023|Discrete Mathematics]] &bull; </li><li style="display:inline-block; padding:2px">[[Calculus_DSBA_2022/2023|Calculus]] &bull; </li><li style="display:inline-block; padding:2px">[[Introduction_to_Programming_DSBA_2022/2023|Introduction to Programming]] &bull; </li><li style="display:inline-block; padding:2px">[[DSBA_Algebra_2022_2023|Algebra]] </li></ul>
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  </div></td>
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</tr>
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</table>
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</div>

Текущая версия на 22:57, 16 января 2024

Teachers and assistants

Группа 221 222 223 224
Lecturer Dima Trushin Telegram
Teacher Dima Trushin Andrew Mazhuga Nikita Medved Galina Kaleeva
Assistant Misha Gorodov Dasha Ivanova Artem Makarenkov

Alena Chislova

Consultations schedule

Teacher/Assistant How to contact When
1
 Dima Trushin telegram Write me and we will schedule a meeting
2
 Andrew Mazhuga telegram
3
 Nikita Medved
4
 Galina Kaleeva Tuesday, 18:00, zoom. Please notify me beforehand
5
 Misha Gorodov telegram
6
 Dasha Ivanova telegram
7
 Artem Makarenkov telegram
8
 Alena Chislova telegram Write me and we will schedule a meeting

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 6 (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.

Lecture 7 (25.05.2023). 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 (01.06.2023). 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 (08.06.2023). 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 (15.06.2023). The Diamond Lemma. A proof of the Buchberger criterion. The Dickson Lemma and termination of the Buchberger algorithm.

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

Seminar 6 (18.05.2023). Problems

Seminar 7 (25.05.2023). Problems

Seminar 8 (01.06.2023). Problems

Seminar 9 (08.06.2023). Problems

Test

The test will take place on Monday 19 of June, since 10:00 in online format. The following file contains all the information.

Exam

The exam will take place on June 24, Saturday.

  • List of definitions and statements.
  • List of statements to prove.
  • The rules for the exam.

The schedule for the exam. You must come at the time in the schedule.

Results

  • Homework
221 222 223 224
  • Test
221 222 223 224
  • Summary Statement
221 222 223 224

Links

  • Telegram chat of the course.

Navigation

DSBA 2022/2023
First year
Источник — «http://wiki.cs.hse.ru/index.php?title=DSBA_Algebra_2022_2023&oldid=82947»