Algebra (Math 200 B)

Winter 2019

 Lectures: T, Th 9:30 AM--10:50 AM APM 5402 Office Hour: T, Th 11:00 AM--12:00 PM APM 7230
 TA: Francois Thilmany (fthilmanucsd edu) Office hour: T 2:30 PM--4:30 PM APM 6132 or APM 6218

Book

Here is a list of good textbooks that you can consult for reference. I will be posting my lecture notes in this page, and I hope you would find them useful. I will not cover the topics in the same order as they are presented in the main text book; and we will learn some topics that are not covered in the main book. The main book has an excellent collection of problems and examples; and I encourage you to go over them and ask me or Francois if you have difficulty solving any of them.

• Dummit and Foote, Abstract Algebra third edition. (The main text book).
• Isaacs, Algebra .
• Hungerford, Algebra .
• Morandi, Field and Galois theory . (For the second half of the course).

Topics

This is a continuation of math 200 a. In this course we continue the study of rings and then move to module theory. The second half of the course is devoted to field and Galois theory. Our goal is to cover all the main topics covered in the first 14 chapters of Dummit and Foote's text.

Assignments.

Problem sets will be posted here. Make sure to refresh your bowser.

• Due Jan 17: Here is the first problem set.
• Due Jan 24: Here is the second problem set.
• Due Feb 7: Here is the third problem set.
• Due Feb 21: Here is the fourth problem set.
• Due Mar 7: Here is the fifth problem set.
• Due Mar 14: Here is the sixth problem set.
• Not due: Here is the seventh problem set.
My notes.

I will post my notes here. You are supposed to read these notes and the relevant sections of your book.

• Lecture 1: Here is my note for the first lecture.

We proved $$D$$ is a UFD if and only if $$D[x]$$ is a UFD. Along the way we proved the following results: Let $$D$$ is a UFD and $$F$$ be its field of fractions. Then $$d\in D$$ is irreducible in $$D$$ if and only if $$d$$ is irreducible in $$D[x]$$; A primitive polynomial $$f(x)\in D[x]$$ of positive degree is irreducible in $$D[x]$$ if and only if it is irreducible in $$F[x]$$. We also mentioned an irreducibility criterion.

• Lecture 2: Here is my note for the second lecture.

We proved Eisenstein's criterion. We defined the ring of fractions with respect to a multiplicative set, and proved its universal property. We started module theory; mentioned that as group actions are crucial in understanding groups, modules are fundamental in understanding rings. Along the way, the opposite ring was defined; mentioned that modules can be viewed as a generalization of vector spaces; left ideals are modules; $$R^n$$ is an $${\rm M}_n(R)$$-module; if $$M$$ is a $$S$$-module and $$f:R\rightarrow S$$ is a unital ring homomorphism, then $$M$$ can be viewed as an $$R$$-module.

• Lecture 4: Here is my note for the fourth lecture.

• Lecture 7: Here is my note for the seventh lecture.

• Lecture 8: Here is my note for the eighth lecture.

• Lecture 9: Here is my note for the ninth lecture.

• Lecture 10: Here is my note for the tenth lecture.

• Lecture 11: Here is my note for the eleventh lecture (short lecture because of midterm).

• Lecture 12: Here is my note for the twelfth lecture.

• Lecture 13: Here is my note for the thirteenth lecture.

• Lecture 14: Here is my note for the fourteenth lecture.

• Lecture 15: Here is my note for the fifteenth lecture.

• Lecture 16: Here is my note for the sixteenth lecture.

• Lecture 17: Here is my note for the seventeenth lecture.

• Lecture 18: Here is my note for the eighteenth lecture.

• Lecture 19: Here is my note for the nineteenth lecture.

• Lecture 20: Here is my note for the twentieth lecture.

Homework

• Homework are due on Thursdays 9:45 am.
• Late Homework are not accepted.
• There will be 8 problem sets. Your cumulative homework grade will be based on the best 7 of the 8.
• You can work on the problems with your classmates and discuss them with me or Francois, but you have to write down your own version. Copying from other's solutions is not accepted and is considered cheating. Copying from an online solution bank is not acceptable, either.
• Only selected problems will be scored, but you are responsible for understanding all the posted problems (including the ones that are not part of the homework assignments).
• A good portion of the exams will be based on the weekly problem sets. So it is extremely important for you to make sure that you understand each one of them.

• Your weighted score is the best of
• Homework 25%+ midterm 25%+ Final 50%
• Homework 25%+ Final 75%
• Your letter grade is determined by your weighted score. And it is meant to suggest how your current performance corresponds to your likely result on the qualifying exam to be held next year: A = PhD Pass, A- = Provisional PhD Pass, B+/B = Masterâ€™s Pass, C or less = not likely to pass the qual.
Further information
• There is no make-up exam.
• Keep all of your returned homework and exams. If there is any mistake in the recording of your scores, you will need the original assignment in order for us to make a change.
• No notes, textbooks, calculators and electronic devices are allowed during exams.
Exams.

• Midterm:
• Time: Tuesday, Feb 12, 2019.
• Location: this is an in-class exam.
• Topics: Till end of split short exact sequences; before basics of category theory.
• Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve anyone of them.
• Exam:
• The final exam:
• Date: Tuesday, Mar 19, 2019, 8:00a-10:59a.
• Location: APM 5402.
• Topics: All the topics that were discussed in class, your homework assignments, and relevant examples and exercises in your book.
• Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve anyone of them.
• Exam: