Modern Algebra II (Math 103 B)

Winter 2019

 Lectures: T-Th 12:30 PM--1:50 PM HSS 1330 Office Hour: T-Th 2:00 PM--3:00 PM APM 7230
 Discussion: W 5:00 AM--5:50 AM APM 7421 TA: John Yin (jbyinucsd edu) Office hour: M 1-3pm APM 2313
 Discussion: W 6:00 PM--6:50 PM APM 7421 TA: Venkata Tunuguntula (tkarthikucsd edu) Office hour: T 5-6p 2202 APM
 Discussion: W 7:00 PM--7:50 PM APM 7421 TA: Venkata Tunuguntula (tkarthikucsd edu) Office hour: Th 4-5p 2202 APM
 Discussion: W 8:00 PM--8:50 PM APM 7421 TA: Itai Maimon (imaimonucsd edu) Office hour: M 4:15-5:15p Location

Book

• John B. Fraleigh, A first course in Abstract Algebra seventh edition. (The main text book).
• Lecture notes will be posted in this webpage.
• J. A. Gallian, Contemporary abstract algebra.

Schedule

Here is a tentative schedule.

Assignments.

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

• Due Jan 17: Here is the first problem set.

The following problems are not part of the problem set, but you have to know how to solve them for the exams.

• Section 18, problems 1, 3, 6, 8, 19, 20, 21, 22, 23, 24, 25, 27, 28, 32, 38, 40, 44.

• Due Jan 24: Here is the second problem set.

The following problems are not part of the problem set, but you have to know how to solve them for the exams.

• Section 19, problems 2, 3, 10, 12, 14, 30.
• Section 20, problem 27.

• Due Feb 7: Here is the third problem set.

The following problems are not part of the problem set, but you have to know how to solve them for the exams.

• Section 21, problems 1, 2, 17.
• Section 22, problems 4, 7, 11, 16, 17, 25, 27.
• Section 23, problems 1, 2, 3, 4, 6, 8, 9, 10, 11, 26, 34, 35, 36.

• Due Feb 14: Here is the fourth problem set.
• Due Feb 21: Here is the fifth problem set.
• Section 23, problems 12, 14, 16, 17, 18, 19, 21, 22, 30, 31.
• Due Mar 7: Here is the sixth problem set.
• Section 26, probelms 1, 2, 3, 17, 18, 26, 37, 38.
• Section 27, problems 1, 2, 3, 4, 5, 6, 8, 15, 16, 17, 18, 19, 24, 30, 31.
• Section 29, problems 1, 2, 3, 4, 6, 8, 18, 25, 30, 32.
• Due Mar 14: 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.

• Here is the lecture note for the first lecture.

In this lecture, we defined ring, commutative and unital rings, and field . The ring of integers modulo $$n$$ is introduced. The ring of $$n$$-by-$$n$$ matrixes with entries in a ring is introduced. Several examples are mentioned. Basic properties of ring operations were discussed. We defined subring and proved the subring criterion. Ring homomorphism was also defined.

• Here is the lecture note for the second lecture.

In this lecture, first we learned how to use $$\mathbb{Z}_3$$, $$\mathbb{Z}_9$$, and $$\mathbb{Z}_{11}$$ to find the remainder of a division by 3, 9, or 11. Next we defined direct product of finitely many rings; and showed that the direct product of unital rings is a unital ring. We proved that $$c:\mathbb{Z}\rightarrow R, c(n):=n1_R$$ is a ring homomorphism when $$R$$ is a unital ring with identity $$1_R$$. Next we showed that $$c_n:\mathbb{Z}\rightarrow \mathbb{Z}_n, c_n(a):=$$ remainder of $$a$$ divided by $$n$$ is a ring homomorphism. Then we defined $$c_{n,m}:\mathbb{Z}_n \rightarrow \mathbb{Z}_m$$ in a similar way and explored if it is a ring homomorphism; and if not under what conditions it is. It turns out that in general it is not; but if $$m|n$$, it is. We did not finish prove of this statement; but we showed that $$c_{n,m}(c_n(a))=c_m(a)$$; along the way we said what it means to say a diagram is commuting .

• Here is the lecture note for the third lecture.

• Here is the lecture note for the fourth lecture.

• Here is the lecture note for the fifth lecture.

• Here is the lecture note for the sixth lecture.

• Here is the lecture note for the seventh lecture.

• Here is the lecture note for the eighth lecture.

• Here is the lecture note for the ninth lecture.

• Here is the lecture note for the tenth lecture.

• Here is the lecture note for the eleventh lecture.

• Here is the lecture note for the twelfth lecture.

• Here is the lecture note for the thirteenth lecture.

• Here is the lecture note for the fourteenth lecture.

• Here is the lecture note for the fifteenth lecture.

• Here is the lecture note for the sixteenth lecture.

• Here is the lecture note for the seventeenth lecture.

• Here is the lecture note for the eighteenth lecture.

• Here is the lecture note for the nineteenth lecture.

• Here is the lecture note for the twentieth lecture.

Homework

• Homework are due on Thursdays. You should hand them to me after the lecture.
• Late Homework are not accepted.
• There will be 9 problem sets. Your cumulative homework grade will be based on the best 8 of the 9.
• You can work on the problems with your classmates, but you have to write down your own version. Copying from other's solutions is not accepted and is considered cheating.
• 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 20%+ exam I 20%+ exam II 20%+ Final 40%
• Homework 20%+ The best of exams I and II 20%+ Final 60%
• Your letter grade is determined by your weighted score using the best of the following methods:
•  A+ A A- B+ B B- C+ C C- D F 97 93 90 87 83 80 77 73 70 67 67>
•  Based on a curve where the median corresponds to the cut-off B-/C+.
• Homework and midterm exams will be returned in the discussion sections.
• If you wish to have your homework or exam regraded, you must return it immediately to your TA.
• Regrade requests will not be considered once the homework or exam leaves the room.
• If you do not retrieve your homework or exam during discussion section, you must arrange to pick it up from your TA within one week after it was returned in order for any regrade request to be considered.
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.

• The first exam:
• Time: Jan 29
• Location: this is an in-class exam. (the second half of the lecture).
• Topics: Briefly here are the topics that we discussed during lectures:
• Remainder divided 3, 9, and 11.
• What a ring homomorphism and isomorphism are; $$c:\mathbb{Z}\rightarrow R, c(n):=n 1_R$$ is a ring homomorphism if $$R$$ is a unital ring.
• Natural homomorphisms between $$\mathbb{Z}_n$$'s; Chinese remainder theorem.
• Euler’s phi function and why it is a multiplicative function.
• Field implies integral domain.
• Finite+integral domain implies field.
• $$\mathbb{Z}_n$$ is a field if and only if it is an integral domain if and only if $$n$$ is prime.
• Field of fractions and its universal property.
• How to use the universal property of field of fractions to show for instance why $$\mathbb{Q}[i]$$ is the field of fractions of $$\mathbb{Z}[i]$$.
• Ring of polynomials; the degree function.
• Assuming $$D$$ is an integral domain, we have $$\deg fg=\deg f+\deg g$$, $$D[x]$$ is an integral domain, and $$D[x]^{\times}=D^{\times}$$.
• $$1-2x$$ is a unit in $$\mathbb{Z}_8[x]$$; more generally $$1-a$$ is a unit if $$a$$ is nilpotent.
• Long division in $$F[x]$$ when $$F$$ is a field. We proved the existence part.
• 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 second exam:
• Time: Feb 26
• Location: this is an in-class exam. (the second half of the lecture).
• Topics: All the topics that are discussed in lectures till the end of lecture 13.
• 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: Mar 19, 11:30-2:30.
• Location: TBA
• 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.