| Time | Mon, Wed 1:10–2:25pm |
| Place | Room 520, Math building |
| Instructor | Anand Deopurkar (anandrd at math) |
| TAs | Andrea Heyman (heyman at math) |
| Office hours |
Anand: Tue, Wed 5:30–6:30pm in Math 413 + MWF 5:30–6:30pm during reading/exam weeks. Andrea: Tue 2:40–4:00pm in Math 520 |
| Textbook | Algebra (2nd edition) by Michael Artin |
| Prerequisites | Modern Algebra I or the instructor's explicit permission |
Congratulations on finishing the Modern Algebra sequence. I hope you enjoyed the course and learned a lot.
The final exam will be on Tuesday, 05/13/2014 from 1:10pm to 4:00pm in Math 520. It will be cumulative, but with a bias towards fields and Galois theory. To prepare, go through the notes/book, make sure you remember all the definitions, and know the precise statements, and proofs of the theorems. Also make sure you can do the homework problems. In addition, here are some additional resources:
Problem 6 on the final practice problem is quite hard. I have outlined a solution here.
.Here is how to solve the cubic.
I have written a general outline of our approach of Galois theory, highlighting the important intermediate steps.
In the first problem, take "Galois theory" in a loose sense. What is useful here is not the main theorem of Galois theory itself, but the general idea of using field automorphisms.
We are following Dummit and Foote's Chapter 13 and 14 for Galois theory, abbreviated by DF in the readings. In class on April 8/10, we mostly did DF 13.4, 13.5, and 14.1 with the omission of algebraic closures in 13.4 and the details of inseparable polynomials in 13.5. On the other hand, we proved the inequality in Proposition 5 in 14.1 for all finite extensions, following their suggestion.
The second midterm will be on Thursday, April 3. For preparation, make sure you know the definitions and can prove the statements from class. Also make sure you know how to do the homework problems. Here is a list of practice problems for you to try.
For a review of vector spaces, see Chapter 3, sections 3 and 4 of the textbook.
Here are solutions of midterm 1.
Pictures of Gaussian primes and Eisenstein primes.
We have finished grading the first midterm. The scores are available on CourseWorks. The mean and median were around 36 (out of 60) and the standard deviation was about 12. If you did poorly, please do not hesitate to come and seek help. We can meet during office hours or schedule alternate times.
The first midterm will be on Thursday, February 20. It will cover everything up to the material on Thursday, February 13. You should know the statements and proofs of all the propositions we proved in class. Also make sure you know how to do the homework problems. Here is a list of practice problems for you to try.
Here are the notes for the class on February 13 for those of you who missed it because of the storm.
Although class is not cancelled today, I will post detailed notes of the material covered.
The statement of problem 3 on homework 2 was wrong! We need an additional assumption that the ring R is a domain. I have added a clarifying remark. I am sorry for being careless. Thanks Irit for pointing it out.
Andrea will hold office hours in the same room (Math 520) on Tuesdays after class from 2:40 to 4:00. These are in addition to my office hours on Tuesday and Wednesday from 5:30 to 6:30.
This is a tentative plan of the course.
| Week | Topic | Reading | Homework |
| Jan 21, 23 |
Introduction Rings and fields Polynomial rings Homomorphisms and ideals |
11.1, 11.2, 11.3 | Homework 1 Solutions |
| Jan 28, 30 |
Quotient rings Application: adding relations and adjoining elements |
11.4, 11.5, 11.6 | Homework 2 Solutions |
| Feb 4, 6 |
Integral domains Fraction fields Prime ideals and maximal ideals |
11.7, 11.8 | Homework 3 Solutions |
| Feb 11, 13 |
Maximal ideals of C[x] Factorization, ED, PID, UFD |
15.10, 12.1, 12.2 | Homework 4 Solutions |
| Feb 18, 20 |
Review Midterm 1 |
||
| Feb 25 | Last day to drop for most schools | ||
| Feb 25, 27 |
Gaussian integers Eisenstein integers |
12.5 | |
| Mar 4, 6 | Factorization in Z[x] Fields |
12.3, 12.4, 15.1 | Homework 5 Solutions |
| Mar 11, 13 |
Fields, Field extensions Review of linear algebra over a field (3.3, 3.4) Degree of a field extension |
15.2, 15.3 | Homework 6 Solutions |
| Mar 17 – 21 | Spring break | ||
| Mar 25, 27 |
Ruler and compass constructions Degree two extensions |
15.5, 15.6, 15.7 | |
| Apr 1, 3 |
Finite fields Algebraically closed fields Midterm 2 |
15.10 | Homework 7 Solutions |
| Apr 8, 10 |
Splitting fields Repeated roots, separability Field automorphisms, fixed fields |
DF 13.4, 13.5, 14.1 | Homework 8 Solutions |
| Apr 15, 17 |
Galois extensions Main theorem of Galois theory |
DF 14.2 | Homework 9 Solutions |
| Apr 22, 24 | Cubic, quartic equations Roots of unity Elementary symmetric functions |
16.8, 16.9, 16.10, 16.1 | Homework 10 Solutions |
| Apr 29, May 1 |
Solvability Insolvability of the quintic |
16.1, 16.12 |
The goal of this course is to study the theory of rings and fields, which can be thought of as structures with two binary operations resembling addition and multiplication. Our usual number systems, such as Z, Q, R, and C are obvious examples, but there are several others. After setting up the basic theory, we will learn to do number theory in these new structures, generalizing the notions of prime numbers, division algorithm, unique factorization, and so on. Using the ideas of rings and fields, we will then study solutions of polynomial equations. In particular, we will see how solutions of polynomial equations have inherent symmetries, and how these symmetries can be used to answer questions about the solutions. The cherry on top will be solutions to the following classical questions:
This course is a continuation of Modern Algebra I. The webpage for Modern Algebra I (Fall 2013) is here.
The final grade will be based on homework (20%), two in-class midterms (20% each), and a final (40%). The exams are scheduled as follows:
It is impossible to learn algebra without working through examples, so the homework forms an essential part of the course. It will be announced weekly on the website and will be due in a week (usually on Wednesday). No late homeworks will be accepted. To compensate, I will drop the two lowest homework scores.
Collaboration: In a mathematics course at this level, you learn as much from your peers as from your instructor. Therefore, you are encouraged to work with each other. However, make sure that you give the problems a serious try before you ask for help. You must write up the solution in your own words. Do not copy anyone else's written work. As a matter of academic honesty, write the names of your collaborators on top of your submission (this will not affect your grade).
