Modern Algebra II (Math 103 B)

Winter 2020

 Lectures: T-Th 11:00 AM--12:20 PM PCYNH 122 Office Hour: W 11:00 AM--1:00 PM APM 7230
 Discussion: W 4:00 PM--4:50 PM APM 2402 TA: Shubham Sinha (shs074ucsd edu) Office hour: Day Time Location
 Discussion: W 5:00 PM--5:50 PM APM 2402 TA: Shubham Sinha (shs074ucsd edu) Office hour: Day Time 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.

Schedule

• This is summary what we have covered in this course.
Homework

• Homework will be assigned in the assignment section of this page.
• Homework are due on Thursdays at 5:00 pm. You should drop your homework assignment in the homework drop-box in the basement of the APM building
• Late homework is not accepted.
• There will be 8 problem sets. Your cumulative homework grade will be based on the best 7 of the 8.
• Selected problems on the each assignment will be graded.
• Style:
• A messy and disorganized homework might get no points.
• The upper right corner of each assignment must include:
1. Your name (last name first).
2. Your discussion session (e.g. A01,etc.).
3. Homework assignment number.
• Full-sized notebook papers should be used.
• All pages should be stapled together.
• Problems should be written in the same order as the assignment list. Omitted problems should still appear in the correct order.
• As a math major, sooner or later you have to learn how to use LaTex. I really encourage you to use Latex to type your solutions.
• 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.
• 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.
• Reading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment. You are responsible for material in the assigned reading whether or not it is discussed in the lecture.
• Homework will be returned in the discussion sections.

• Your weighted score is the best of
• Homework 20%+ midterm exam I 20%+ midterm exam II 20%+ Final 40%
• Homework 20%+ The best of midterm exams 20%+ Final 60%
• You must pass the final examination in order to pass the course.
• 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- 97 93 90 87 83 80 77 73 70
•  Based on a curve where the median corresponds to the cut-off B-/C+.
• If more than 90% of the students fill out the CAPE questioner at the end of the quarter, all the students get one additional point towards their weighted score.
• 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.
• Academic Dishonesty: Academic dishonesty is considered a serious offense at UCSD. Students caught cheating will face an administrative sanction which may include suspension or expulsion from the university. It is in your best interest to maintain your academic integrity.
Exams.

• The first exam:
• Time: Wednesday, January 29, 20:00-20:50.
• Location: CSB001
• Topics: TBA
• 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.
• Extra office hour: Monday, January 27, 9a-10a.
• Practice: besides going through your homework assignments, examples presented in the class and problems in the relevant chapters of your book, you can use the following practice exams:
• The second exam:
• Time: Wednesday, February 26, 20:00-20:50.
• Location: CSB001
• Topics: All the topics that are discussed in class and Sections 18, 19, 21, 22, 23, and 26. Make sure that you know all the topics covered in the lecture notes; for instance the fact that $$a^p=a$$ in $$\mathbb{Z}_p$$; if characteristic of a unital commutative ring $$R$$ is a prime $$p$$, then $$F_p:R\rightarrow R, F_p(a)=a^p$$ is a ring homomorphism; binomial expansion; if $$F$$ is a field, then $$F[x]$$ is a PID; and Gauss's lemma.
• 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.
• Extra office hour: Monday, February 24, 9a-10a.
• Practice: go through your homework assignments, examples presented in the class and problems in the relevant chapters of your book. You can find practice exams here:
• The final exam:
• Time: Thursday, March 19, 11:30-14:30.
• You will be provided an electronic version of the exam. Here is the front page of the exam.
• The exam will be proctored by Zoom. A Zoom invitation will be sent to you for the exam; it will be schedualed for Thursday 11:15 am.
• You should prepare 10-15 empty sheets of paper for the exam; you will be provided with a one-page exam sheet that contains only questions with no space for writing solutions.
• After you are done, you should let me know through Zoom, take pictures of your exam sheets and/or scan them and post them in gradescope. Here you can find the instruction of posting pictures of exams in gradescope.
• Your exam should be posted in gradscope no later than 15:00pm.
• If/when you have a question during the exam, you can send a message to me or your TA via Zoom.
• It is your responsibility to ensure that you do not have a schedule conflict involving the final examination. You should not enroll in this class if you cannot sit for the final examination at its scheduled time.
• Topics: All the topics that are discussed in the class and in the book. My lecture notes here, here and here can be useful. I have prepared a summary of lectures ; it should be useful.
• 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.
• Extra office hour: There will be two office hours through Zoom; invitations will be sent later. One on Monday 11-12, and the other on Wednesday 11-12. Even if you do not have any math question, please try the Zoom link and make sure that you know how to work with Zoom.
• Practice: go through your homework assignments, examples presented in the class and problems in the relevant chapters of your book. You can find practice exams here:
• As it is stated above, in the worst case scenario the median of the weighted scores corresponds to the B-/C+ cut-off.
• Lectures
I am closely following my lectures from Summer 2017; here you can find the links to the lecture notes that I had prepared at that time. These lecture notes can only be a complement to the notes that you take during the lectures. Notice that your book is an excellent source and you should study the examples presented in your book; we, however, cover some topics in more depth. For instance, we discuss Gauss's lemma and prove some theorems that are merely stated in your book; so make sure that you attend the lectures.
Assignments
• Homework 1 (Due 01-23)
• Chapter 18. Problems 12, 18, 26, 28, 38
• Chapter 19. Problems 1, 17(a-h).
• Solutions are provided by your TA pdf.
The following problems are not part of the problem set, but you have to know how to solve them for the exams.
• Chapter 18, problems 1, 3, 6, 8, 19, 20, 21, 22, 23, 24, 25, 27, 32, 40, 44.
• Chapter 19, problems 2, 3, 10, 12, 14, 30.

• Homework 2 (Due 02-06)
• Chapter 19. Problems 24, 27, 28.
• Chapter 21. Problems 2, 4(a-e,h-j), 5.
• Chapter 22. Problems 5, 8, 11, 25(b,c).
• Solutions are provided by your TA pdf.
The following problems are not part of the problem set, but you have to know how to solve them for the exams.
• Chapter 21. Problems 1, 12, 17.
• Chapter 22, Problems 4, 7, 16, 17, 27.

• Homework 3 (Due 02-20 )
• These problems PDF . (We have proved the first problem in class; go over your notes and reproduce that proof.)
• Chapter 22. 17.
• Chapter 23. 34, 37(c) (Hint: use $$p=5$$).
• Solutions are provided by your TA pdf.
The following problems are not part of the problem set, but you have to know how to solve them for the exams.
• Chapter 22. 11, 15, 13, 16, 25, 27.
• Chapter 23. 1, 2, 3, 4, 6, 8, 9, 10, 11, 14, 16, 17, 18, 19, 21, 22, 26, 30, 31, 35, 36.

• Homework 4 (Due 02-27 )
• 1. Suppose $$E$$ is a finite integral domain of characteristic $$p$$. Let $$F_p:E\rightarrow E, F_p(x):=x^p$$. Prove that $$F_p$$ is a ring isomorphism. (Long ago in class we proved that $$F_p$$ is a ring homomorphism in any ring of characteristic $$p$$ when $$p$$ is prime. Go over your notes and rewrite that part of the argument as well. Notice that you have to argue why $$p$$ is prime and why $$F_p$$ is a bijection.)
• 2.
• (a) Prove that the minimal polynomial of $$\alpha:=\sqrt{1+\sqrt{3}}$$ is $$x^4-2x^2-2$$.
• (b) Prove that $$\mathbb{Q}[\alpha]:=\{c_0+c_1\alpha+c_2\alpha^2+c_3\alpha^3|c_0,c_1,c_2,c_3\in\mathbb{Q}\}$$ is a subring of $$\mathbb{C}$$.
• (c) Prove that $$\mathbb{Q}[x]/\langle x^4-2x^2-2\rangle \simeq \mathbb{Q}[\alpha]$$.
• (d) Write $$\alpha^{-1}$$ in the form $$c_0+c_1\alpha+c_2\alpha^2+c_3\alpha^3$$ with $$c_i\in \mathbb{Q}$$.
• (e) Write $$(1+\alpha)^{-1}$$ in the form $$c_0+c_1\alpha+c_2\alpha^2+c_3\alpha^3$$ with $$c_i\in \mathbb{Q}$$.
• 3. Suppose $$E$$ is a finite field that contains $$\mathbb{Z}_3$$ as a subring. Suppose there is $$\alpha\in E$$ such that $$\alpha^3-\alpha+1=0$$. Let $$\phi_\alpha:\mathbb{Z}_3[x]\rightarrow E$$ be the map of evaluation at $$\alpha$$.
• (a) Prove that $$\ker \phi_\alpha=\langle x^3-x+1\rangle$$.
• (b) Prove that $${\rm Im} \phi_\alpha=\{c_0+c_1\alpha+c_2\alpha^2|\hspace{1mm} c_0,c_1,c_2\in \mathbb{Z}_3\}$$.
• (c) Let us denote the image of $$\phi_\alpha$$ by $$\mathbb{Z}_3[\alpha]$$. Prove that $$\mathbb{Z}_3[\alpha]$$ is a finite field with $$27$$ elements.
• 4. Suppose $$I$$ and $$J$$ are two ideals of a commutative ring $$R$$.
• (a) Prove that $$I\cap J$$ is an ideal of $$R$$.
• (b) Let $$I+J:=\{x+y|\hspace{1mm} x\in I, y\in J\}$$. Prove that $$I+J$$ is an ideal of $$R$$.
• 5. Suppose $$R$$ is a unital commutative ring and $$x_1,\ldots,x_n\in R$$.
• (a) Recall that $$\langle x_i\rangle=Rx_i=\{rx_i|\hspace{1mm} r\in R\}$$ the ideal generated by $$x_i$$. Let $$I:=Rx_1+Rx_2+\cdots+Rx_n=\{r_1x_1+\cdots+r_nx_n|\hspace{1mm} r_1,\ldots,r_n\in R\}$$. Prove that $$I$$ is an ideal.
• (b) Prove that the ideal $$I$$ given in part (a) is the smallest ideal that contains $$x_1,\ldots,x_n$$; we say $$I$$ is generated by $$x_1,\ldots,x_n$$.
• 4. Let $$I:=\langle 2,x\rangle=\{2f(x)+x g(x)|\hspace{1mm} f(x),g(x)\in \mathbb{Z}[x]\}$$. Prove that $$I$$ is not a principal ideal. Deduce that $$\mathbb{Z}[x]$$ is not a PID.
• 6. (This is a hard problem, and questions similar to this will not be in your exams; but this result was mentioned earlier in class. I believe each step is doable.) Suppose $$E$$ is a finite field that contains $$\mathbb{Z}_p$$ as a subring. Suppose $$a\in \mathbb{Z}_p^{\times}$$. Suppose there is $$\alpha\in E$$ such that $$\alpha^p-\alpha+a=0$$.
• (a) Prove that $$\alpha+1, \alpha+2, \ldots,\alpha+(p-1)$$ are zeros of $$x^p-x+a$$.
• (b) Prove that in $$E[x]$$ we have $$x^p-x+a=(x-\alpha)(x-\alpha-1)\cdots(x-\alpha-p+1)$$. (Use the generalized factor theorem and compare the leading coefficients.)
• (c) Suppose $$f(x)$$ is a divisor of $$x^p-x+a$$. Argue why $$f(x)=(x-\alpha-i_1)\cdots (x-\alpha-i_d)$$ for some $$i_1,\ldots,i_d\in \mathbb{Z}_p$$.
• (d) Argue why $$f(x)=(x-\alpha-i_1)\cdots (x-\alpha-i_d)$$ implies that the coefficient of $$x^{d-1}$$ of $$f$$ is $$-(d\alpha+i_1+\cdots+i_d)$$.
• (e) Suppose $$f(x)\in \mathbb{Z}_p[x]$$ is a divisor of $$x^p-x+a$$ and $$\deg f< p$$. Prove that $$\alpha\in \mathbb{Z}_p$$.
• (f) Use the previous part and Fermat's little theorem to get a contradiction, and deduce that $$x^p-x+a$$ is irreducible in $$\mathbb{Z}_p[x]$$.
• The following problems are not part of the problem set, but you have to know how to solve them for the exams.
• Chapter 26. 1, 2, 3, 18, 26, 30, 37, 38.
• Chapter 27. 32, 35, 36.
• Solutions are provided by your TA pdf.
• Last recommended set of problems The following problems are not part of the problem set, but you have to know how to solve them for the exams.
• Chapter 27. 5, 6, 8, 24.
• Chapter 29. 4, 5, 25, 30, 34, 36.
• Chapter 33. 10, 12.