Algebra DSBA 2019/2020 — различия между версиями
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(не показаны 43 промежуточные версии 3 участников) | |||
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+ | = Schedule = | ||
+ | |||
+ | * [https://zoom.us/j/98706972969 Lecture] Monday 12:10–13:30 | ||
+ | |||
+ | * [https://zoom.us/j/92828388875 Seminar 191] Monday 13:40–15:00 | ||
+ | |||
= Teachers and assistants = | = Teachers and assistants = | ||
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! !! Teacher/Assistant !! Monday !! Tuesday !! Wednesday !! Thursday !! Friday | ! !! Teacher/Assistant !! Monday !! Tuesday !! Wednesday !! Thursday !! Friday | ||
|- | |- | ||
− | | <center>1</center> || Dima Trushin || || || | + | | <center>1</center> || Dima Trushin || || || [https://us04web.zoom.us/j/5515049969 zoom since 16:00] || || |
|- | |- | ||
− | | <center>2</center> || Sergey Gayfullin || || | + | | <center>2</center> || Sergey Gayfullin || || [https://us02web.zoom.us/j/89143777345 16:30–18:00] || || || |
|- | |- | ||
− | | <center>3</center> || Galina Kaleeva || | + | | <center>3</center> || Galina Kaleeva || [https://us04web.zoom.us/j/75909851166?pwd=MzQ2MmpQZTFiaUs5cEd0c1NxekpSdz09 16:30–18:00] пароль: algebra|| || || || [https://us04web.zoom.us/j/75909851166?pwd=MzQ2MmpQZTFiaUs5cEd0c1NxekpSdz09 16:40–18:00] (June, 5th) пароль: algebra |
|- | |- | ||
| <center>4</center> || Arina || || || || || | | <center>4</center> || Arina || || || || || | ||
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= Lecture abstracts = | = Lecture abstracts = | ||
+ | |||
+ | '''Lecture 1''' (06.04.2020). 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''' (13.04.2020). Subgroups of the group of integers. Left and right cosets, examples. Normal subgroups. The Lagrange theorem and its 5 corollaries. | ||
+ | |||
+ | '''Lecture 3''' (20.04.2020). 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''' (27.04.2020). 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''' (07.05.2020). 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''' (12.05.2020). 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''' (18.05.2020). 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''' (25.05.2020). 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. | ||
+ | |||
+ | '''Lecture 9''' (01.06.2020). Stabilization of reduction. 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. | ||
= Problem sheets = | = Problem sheets = | ||
+ | |||
+ | The solutions should be sent to your teaching assistant via email before the beginning of the next seminar. | ||
+ | |||
+ | '''Seminar 1''' (06.04.2020). [https://yadi.sk/i/nkiKmPG5Aie7TA '''Problems'''] | ||
+ | |||
+ | '''Seminar 2''' (13.04.2020). [https://yadi.sk/i/mcXh-_s6xpcCMA '''Problems'''] | ||
+ | |||
+ | '''Seminar 3''' (20.04.2020). [https://yadi.sk/i/UkSrO9JhNa14sg '''Problems'''] | ||
+ | |||
+ | '''Seminar 4''' (27.04.2020). [https://yadi.sk/i/A9JjBjJfsHL9_Q '''Problems'''] | ||
+ | |||
+ | '''Seminar 5''' (07.05.2020). [https://yadi.sk/i/xWV0DbYfPG9qiQ '''Problems'''] | ||
+ | |||
+ | '''Seminar 6''' (12.05.2020). [https://yadi.sk/i/jdg96r9XChrFkA '''Problems'''] | ||
+ | |||
+ | '''Seminar 7''' (18.05.2020). [https://yadi.sk/i/JzwF9Rd5Ycn6dA '''Problems'''] | ||
+ | |||
+ | '''Seminar 8''' (25.05.2020). [https://yadi.sk/i/qWymXdWqGB9KzA '''Problems'''] | ||
+ | |||
+ | '''Seminar 9''' (01.06.2020). [https://yadi.sk/i/H0QLPDojzbQVXQ '''Problems'''] | ||
+ | |||
+ | = Exam = | ||
+ | |||
+ | * [https://yadi.sk/i/oY_3ADtOrqxWww The lists of definitions and statements] | ||
+ | |||
+ | * [https://yadi.sk/i/Zb7RvwpECu8-3A The list of statements to prove] | ||
+ | |||
+ | = Results = | ||
+ | |||
+ | * Homework | ||
+ | |||
+ | {| class="wikitable" style="text-align:center" | ||
+ | |- | ||
+ | ! [https://docs.google.com/spreadsheets/d/1HUJ-WoTtWmpYSNq2okUr99AKX2MmVlC3pnctCMsSEEI/edit#gid=0 191] !! [https://docs.google.com/spreadsheets/d/1HUJ-WoTtWmpYSNq2okUr99AKX2MmVlC3pnctCMsSEEI/edit#gid=1992163396 192] !! [https://docs.google.com/spreadsheets/d/1HUJ-WoTtWmpYSNq2okUr99AKX2MmVlC3pnctCMsSEEI/edit#gid=83275050 193] | ||
+ | |} | ||
+ | |||
+ | * Summary Statement | ||
+ | |||
+ | {| class="wikitable" style="text-align:center" | ||
+ | |- | ||
+ | ! [https://docs.google.com/spreadsheets/d/1ULykRTOgC79vjbaE7hXnVsmEzO2fnWcVwCPrff-IICU/edit#gid=0 191] !! [https://docs.google.com/spreadsheets/d/1ULykRTOgC79vjbaE7hXnVsmEzO2fnWcVwCPrff-IICU/edit#gid=17634846 192] !! [https://docs.google.com/spreadsheets/d/1ULykRTOgC79vjbaE7hXnVsmEzO2fnWcVwCPrff-IICU/edit#gid=1828997773 193] | ||
+ | |} | ||
+ | |||
+ | = Links = | ||
+ | |||
+ | * [https://yadi.sk/d/4WqzAFDXu0iwQw '''Lecture Notes'''] | ||
+ | |||
+ | * [https://yadi.sk/d/UOTiZgeClaHozQ '''Seminars 191'''] |
Текущая версия на 22:12, 14 июня 2020
Содержание
Schedule
- Lecture Monday 12:10–13:30
- Seminar 191 Monday 13:40–15:00
Teachers and assistants
Группа | 191 | 192 | 193 |
---|---|---|---|
Lecturer | Dima Trushin | ||
Teacher | Dima Trushin | Sergey Gayfullin | Galina Kaleeva |
Assistant | Arina | Yunying | Timur |
Consultations schedule
Teacher/Assistant | Monday | Tuesday | Wednesday | Thursday | Friday | |
---|---|---|---|---|---|---|
|
Dima Trushin | zoom since 16:00 | ||||
|
Sergey Gayfullin | 16:30–18:00 | ||||
|
Galina Kaleeva | 16:30–18:00 пароль: algebra | 16:40–18:00 (June, 5th) пароль: algebra | |||
|
Arina | |||||
|
Yunying | |||||
|
Timur |
Grading system
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 = 0,3 * H + 0,2 T + 0,5 E
where 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 (06.04.2020). 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 (13.04.2020). Subgroups of the group of integers. Left and right cosets, examples. Normal subgroups. The Lagrange theorem and its 5 corollaries.
Lecture 3 (20.04.2020). 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 (27.04.2020). 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 (07.05.2020). 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 (12.05.2020). 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 (18.05.2020). 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 (25.05.2020). 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.
Lecture 9 (01.06.2020). Stabilization of reduction. 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.
Problem sheets
The solutions should be sent to your teaching assistant via email before the beginning of the next seminar.
Seminar 1 (06.04.2020). Problems
Seminar 2 (13.04.2020). Problems
Seminar 3 (20.04.2020). Problems
Seminar 4 (27.04.2020). Problems
Seminar 5 (07.05.2020). Problems
Seminar 6 (12.05.2020). Problems
Seminar 7 (18.05.2020). Problems
Seminar 8 (25.05.2020). Problems
Seminar 9 (01.06.2020). Problems
Exam
Results
- Homework
191 | 192 | 193 |
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- Summary Statement
191 | 192 | 193 |
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