Algebraic Structures 2 (80446)

Teacher: Prof. David Kazhdan
Teaching Assistant: Sefi Ladkani

Class hours: Sunday 15-16 (Levi 07), Monday 10-12 (Levi 07)
TA hours: Sunday 11-12 (Levi 07) or Tuesday 14-15 (Shprinzak 102)

TA Reception hour: Tuesday 18-19 (Math 318)

(12.9.05)
Final exam (Moed B) is here. The correct answers are: 1(a), 2(b), 3(b), 4(d), 5(d), 6(a), 7(c), 8(b), 9(c), 10(a).

(4.7.05)
Final grades were computed according to the following formula.

First, the grades with and without Quiz 1 were computed by

Grade_WithQ1 = 0.15 * Grade_HW + 0.1 * Grade_Quiz1 + 0.1 * Grade_Quiz2 + 0.65 * Grade_Exam

Grade_WithoutQ1 = (0.15 * Grade_HW + 0.1 * Grade_Quiz2 + 0.65 * Grade_Exam) / 0.9

and then, the final grade is computed as Grade_final = Max(Grade_WithQ1, Grade_WithoutQ1),
so that the grade of Quiz1 serves as a MAGEN.

(28.6.05)
Updated lists of homework submission (including homework grade) and the grades of the quizzes.

(26.6.05)
Final exam (Moed A) is here. The correct answers are: 1(a), 2(c), 3(b), 4(d), 5(d), 6(a), 7(c), 8(b), 9(b), 10(c).

(22.6.05)
Sample questions for the exam can be found here.

(21.6.05)
Notes on the exam:
As said in the previous lessons, the exam will consist of a reasonable number of multiple-choice questions you will have to answer (no need to prove your answers, no choice).

(21.6.05)
Partial solutions to homeworks 9 and 10 are available.

(21.6.05)
Partial solutions to the second quiz are here.

(20.6.05)
Grades of quiz 1 and 2 can be found here.

(19.6.05)
The lesson tomorrow (Monday, 20.6) at 16:00 will be held in Levi 06 hall.

(16.6.05)
An additional lesson (Q&A) will be held on Monday, 20.6.05 at 16:00. Details on the location will be posted later.

(7.6.05)
Assignment 11 is online.

(31.5.05)
Reception hour is moved from today to Wednesday 1/6 at 17:00 (Math 318).

(31.5.05)
Notes of lecture 12 are online; Assignment 10 is online, due Tuesday 7/6.

(24.5.05)
Notes of lecture 11 are online.

(22.5.05)
Assignment 9 is online.

(18.5.05)
No homework for 22/5.

(17.5.05)
Notes for lecture 10 and solutions for homework 8 are online.
In addition, new version of notes for lecture 8 is available.
New homework will be posted soon.

(8.5.05)
Problems of the take home quiz can be found here. Submission is due on Sunday, May 15, 15:00.

(8.5.05)
Due to Memorial Day eve, the TA reception hour on Tuesday May 10 will be held between 15 to 16, in Shprinzak 102 (after the TA class).

(30.4.05)
Grades of quiz 1 are available here. There is also an explanation on the computation of the grades.

(27.4.05)
Solutions for assignment 5, the first quiz and the sample problems for that quiz are now available.

(18.4.05)
A take-home exam will be given on May 15th.

(18.4.05)
Special reception hour by Prof. Kazhdan: Wednesday 20/4 at 12:00, Room 201 Math building.

(3.4.05)
The grade of the quiz held today will affect 10% of the final grade.

(28.3.05)
Sample problems for the quiz can be found here. Solutions to assignments 1 through 4 are also online.

(24.3.05)
Assignment 5 is due on Sunday 3/4.
Question 5.1 is optional (i.e. you do not have to solve it). Question 5.2 is postponed (it will appear in another assignment).

Course materials

Prerequisites: Good knowledge of linear algebra and the understanding of basic concepts of the group theory.

Topics:

• Fields and fields extensions, elementary extensions, algebraic and transendental extension.
• The degree of an extension and the product formula.
• Ruler and compass constructions impossibility of the trisection of an angle.
• Splitting field and algebraic closure.
• Separable and inseparable extensions.
• Normal extensions and the Galois group.
• Galois theory.
• Construction of perfect n-gons.
• Solvability of equations by radicals.
• Finite fields and Frobenious.
• Fields of positive characteristic.
• Cyclotomic fields and the quadratic reciprocity law.
• Infinite fields extensions and the transcendence degree.

Recommended reading

The following books contain relevant material for the course:

1. Serge Lang, Algebra.
2. Jacobson, Basic algebra.
3. Herstein, Abstract algebra.
4. Postnikov, Teoriia Galua (in Russian).

Lecture 1 (20/2) [pdf]
Lecture 2 (27/2) [pdf]
Lecture 3 (6/3) [pdf] (Note: updated version was uploaded on 13/3)
Lecture 4 (13/3) [pdf]
Lecture 5 (20/3) [pdf]
Lecture 6 (3/4) [pdf] (Note: updated version was uploaded on 5/4)
Lecture 7 (10/4) [pdf]
Lecture 8 (1/5) [pdf] (Note: newest version is of 17/5)
Lecture 9 (8/5) [pdf]
Lecture 10 (15/5) [pdf]
Lecture 11 (22/5) [pdf]
Lecture 12 (29/5) [pdf]

Assignments are very important part of the course. They will contain examples, results and methods not discussed in class. Submission of at least 80 percent of the assignments is mandatory. The grades of the assignments will constitute 15 percent of the final grade in the course.

Assignments are due on the following Sunday. You can hand them in either in class, in the TA class, or in my cell (third floor in Math building).

Assignment 1 [pdf], due 27/2. Solution [pdf]
Assignment 2 [pdf], due 6/3. Solution [pdf]
Assignment 3 [pdf], due 13/3. (Note: an updated version was uploaded on 7/3) Solution [pdf]
Assignment 4 [pdf], due 20/3. Solution [pdf]
Assignment 5 [pdf], due 3/4. (Note: question 5.1 is optional; 5.2 postponed) Solution [pdf]
Assignment 6 [pdf], due 10/4.
Assignment 7 [pdf], due 1/5. Solution [pdf]
Assignment 8 [pdf], due 8/5. (Note: make sure you have the latest version) Solution [pdf]
Assignment 9 [pdf], due 29/5. Solution [pdf]
Assignment 10 [pdf], due 7/6. Solution [pdf]
Assignment 11 [pdf], due 14/6.

Sample problems for Quiz 1 [pdf], solutions [pdf]

Quiz 1 problems [pdf], solutions [pdf]

Sample problems for take-home Quiz 2 [pdf]

Take home Quiz 2 problems [pdf], solutions [pdf] (partial).

Sample problems for the final exam [pdf]