| 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:
-
Summer 2017,
PDF .
-
Winter 2019,
PDF .
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:
-
Summer 2017,
PDF .
-
Winter 2019,
PDF .
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:
-
Summer 2017,
PDF .
-
Winter 2019,
PDF .
- As it is stated above, in the worst case scenario the median of the weighted scores corresponds to
the B-/C+ cut-off.
|
Assignments
The list of homework assignments are subject to revision during the quarter.
Please check this page regularly for updates. (Do not forget to refresh your page!)
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Homework 1 (Due 01-23)
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Chapter 18. Problems 12, 18, 26, 28, 38
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Chapter 19. Problems 1, 17(a-h).
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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.
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Chapter 18, problems 1, 3, 6, 8, 19, 20, 21, 22, 23, 24, 25, 27, 32, 40, 44.
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Chapter 19, problems 2, 3, 10, 12, 14, 30.
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Homework 2 (Due 02-06)
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Chapter 19. Problems 24, 27, 28.
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Chapter 21. Problems 2, 4(a-e,h-j), 5.
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Chapter 22. Problems 5, 8, 11, 25(b,c).
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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.
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Chapter 21. Problems 1, 12, 17.
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Chapter 22, Problems 4, 7, 16, 17, 27.
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Homework 3 (Due 02-20 )
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These problems PDF . (We have proved the first problem in class; go over your notes and reproduce that proof.)
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Chapter 22. 17.
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Chapter 23. 34, 37(c) (Hint: use \(p=5\)).
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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.
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Chapter 22. 11, 15, 13, 16, 25, 27.
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Chapter 23. 1, 2, 3, 4, 6, 8, 9, 10, 11, 14, 16, 17, 18, 19, 21, 22, 26, 30, 31, 35, 36.
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Homework 4 (Due 02-27 )
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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.
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(a) Prove that the minimal polynomial of \(\alpha:=\sqrt{1+\sqrt{3}}\) is \(x^4-2x^2-2\).
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(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 \(\mathbb{C}\).
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(c) Prove that \(\mathbb{Q}[x]/\langle x^4-2x^2-2\rangle \simeq \mathbb{Q}[\alpha]\).
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(d) Write \(\alpha^{-1}\) in the form \(c_0+c_1\alpha+c_2\alpha^2+c_3\alpha^3\) with \(c_i\in \mathbb{Q}\).
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(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 \(\alpha\).
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(a) Prove that \(\ker \phi_\alpha=\langle x^3-x+1\rangle\).
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(b) Prove that \({\rm Im} \phi_\alpha=\{c_0+c_1\alpha+c_2\alpha^2|\hspace{1mm} c_0,c_1,c_2\in \mathbb{Z}_3\}\).
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(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 \(R\).
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(a) Prove that \(I\cap J\) is an ideal of \(R\).
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(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 \(\alpha^p-\alpha+a=0\).
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(a) Prove that \(\alpha+1, \alpha+2, \ldots,\alpha+(p-1)\) are zeros of \(x^p-x+a\).
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(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.)
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(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 \(i_1,\ldots,i_d\in \mathbb{Z}_p\).
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(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 \(-(d\alpha+i_1+\cdots+i_d)\).
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(e) Suppose \(f(x)\in \mathbb{Z}_p[x]\) is a divisor of \(x^p-x+a\) and \(\deg f< p\). Prove that \(\alpha\in \mathbb{Z}_p\).
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(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.
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Chapter 26. 1, 2, 3, 18, 26, 30, 37, 38.
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Chapter 27. 32, 35, 36.
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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.
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Chapter 27. 5, 6, 8, 24.
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Chapter 29. 4, 5, 25, 30, 34, 36.
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Chapter 33. 10, 12.
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