Modern Algebra II (Math 103 B)

Spring 2018

 Lectures: M-W-F 12:00p-12:50p Center 216 Office Hour: M-W-F 1:00p-2:00p APM 7230
 Discussion: Tu 10:00a APM B402A TA: Rosemary Elliott Smith (reelliot ucsd edu) Office hour: M, W 3p-4p APM 5218
 Discussion: Tu 11:00a APM B402A TA: Mingjie Chen (mic181 ucsd edu) Office hour: T 12:30p-2:30p APM 6414
 Discussion: Tu 8:00a APM B402A TA: Mingjie Chen (mic181 ucsd edu) Office hour: T 12:30p-2:30p APM 6414
 Discussion: Tu 9:00a APM B402A TA: Haiyu Huang (hah019 ucsd edu) Office hour: W 2p-3p APM 5218

Book

I will not follow the main text book closely. I will reorder some of the topics as well. But I will post my lecture notes here and I expect you to read my notes and the relavent parts of the main text book.

• J. A. Gallian, Contemporary abstract algebra. Ninth Edition. (Cengage Learning, 2017). (The main text book).
• Lecture notes will be posted in this webpage.
• John B. Fraleigh, A first course in Abstract Algebra.

Schedule

This is a tentative schedule for the course. If necessary, it may change.

Assignments.

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

• Due April 11: Here is the first problem set. Here are their solutions.

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

• Section 12, problems 1, 2, 19, 23, 32, 49, 57.

• Due April 18: Here is the second problem set. Here are their solutions.

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

• Section 13, problems 12, 14, 15, 29, 31, 36.

• Due April 25: Here is the third problem set. Here are their solutions.

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

• Section 14, problems 4, 5, 10, 11, 12, 14.
• Section 15, problems 13, 15, 16, 33, 36, 41, 56.

• Due May 2: Here is the fourth problem set. Here are their solutions.

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

• Section 15, problems 21, 22, 23, 24, 25, 28, 29, 41, 50, 56, 65, 66.

• Due May 9: Here is the fifth problem set. Here are their solutions.

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

• Section 14, problems 13, 18, 19, 27, 29, 32, 34, 55, 60.
• Section 15, problems 37, 47, 65.
• Section 16, problems 9, 61.

• Due May 16: Here is the sixth problem set. Here are their solutions.

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

• Section 16, problems 2, 17, 19, 20, 23, 27, 51, 52.

• Due May 23: Here is the seventh problem set. Here are their solutions.
• Due May 30: Here is the eighth problem set. Here are their solutions.
• Due June 6: Here is the ninth problem set. Here are their solutions.
• Not Due: Here is the tenth 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.

In this lecture, we briefly discussed history of algebra; defined ring; proved basic properties of rings; and mentioned a few examples.

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

In this lecture, we mentioned a subring criterion; defined ring of $$n\times n$$ matrices with entries in a ring $$R$$; defined direct product of finitely many rings; did some computations in $$M_2(\mathbb{Z}\times \mathbb{R})$$; defined zero-divisors; defined the ring of integers modulo $$n$$; recalled basic properties of congruence arithmetic.

• Lecture 3: Here is my note for the third lecture.

In this lecture, we proved arithmetic properties of congruences; showed how to use this to find remainder of dividing by 9 or 11; discussed why $$\mathbb{Z}_n$$ is a ring; wrote the multiplication and addition tables of $$\mathbb{Z}_4$$; observed that 2 is a zero-divisor in $$\mathbb{Z}_4$$; find all the zeros of $$x^2-x$$ in $$\mathbb{Z}_5$$ and $$\mathbb{Z}_6$$; observed that in $$\mathbb{Z}_6$$ this degree 2 polynomial has 4 zeros, and pointed out that this happens because of existence of zero-divisors in $$\mathbb{Z}_6$$. At the end of the lecture we defined a ring homomorphism and isomorphism.

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

In this lecture, we defined characteristic of a ring; showed characteristic of a ring is the least common multiple of the additive order of its elements if the l.c.m. is not infinity, and characteristic is zero if the considered l.c.m. is infinite; when the ring is unital, then its characteristic is the additive order of its unity if $${\rm ord}(1_A)< \infty$$ and it is zero otherwise; proved that $${\rm char}(\oplus_n \mathbb{Z}_n)=0$$ even though any element has finite additive order; defined integral domian and proved that it has the cancellation property.

• Lecture 5: Here is my note for the fifth lecture.

In this lecture, we proved a non-zero unital commutative ring is an integral domain if and only if it has the cancellation property; proved a finite integral domain is a field; proved any field is an integral domain; proved that $$\mathbb{Z}_n$$ is an integral domain if and only if $$n$$ is prime; proved that $$\mathbb{Z}_p$$ is a field if $$p$$ is prime; proved that the characteristic of an integral domain is either zero or a prime; gave the main theorem about the field of fractions of an integral domain.

• Lecture 6: Here is my note for the sixth lecture.

We started proof of existence of a field of fractions of an integral domain. For an integral domain $$D$$, for any $$(a,b)\in D\times (D\setminus \{0\})$$, we let $$[(a,b)]:=\{(c,d)\in D\times (D\setminus \{0\})| ad=bd\}$$. We proved that $$\{[(a,b)]| (a,b)\in D\times (D\setminus \{0\})\}$$ is a partition of $$D\times (D\setminus \{0\})$$. Then we let $$Q(D):=\{[(a,b)]| (a,b)\in D\times (D\setminus \{0\})\}$$ and defined the following binary operators on $$Q(D)$$: $$[(a,b)]+[(c,d)]:=[(ad+bc,bd)]$$ and $$[(a,b)].[(c,d)]:=[(ac,bd)]$$; we proved these are well-defined; we showed that for any $$a\in D\setminus\{0\}$$ we have $$[(0,a)]=[(0,1)]$$ and $$[(1,1)]=[(a,a)]$$, $$[(0,1)]$$ is the neutral element of $$(Q(D),+)$$ and $$[(1,1)]$$ is the unity; and $$(Q(D),+,.)$$ is a ring; next we showed that $$(Q(D),+,.)$$ is a field.

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

For $$\theta: D\rightarrow Q(D), \theta(d):=[(d,1)]$$, we proved that (1) $$\theta$$ is an injective ring homomorphism, $$\theta(1)=1_{Q(D)}$$, and (2) any element of $$Q(D)$$ is of the form $$\theta(a)\theta(b)^{-1}$$ for some $$a\in D$$ and $$b\in D\setminus \{0\}$$. Next we showed that if a field $$Q$$ and a ring homomorphism $$\theta:D\rightarrow Q$$ satisfy (1) and (2) and $$\psi:D\rightarrow F$$ is an injective ring homomorphism into a field $$F$$, then there is an injective ring homomorphism $$\tilde{\psi}:Q\rightarrow F$$ such that $$\tilde{\psi}(\theta(d))=\psi(d)$$ for any $$d\in D$$. Using this, we proved that if $$Q_1$$ and $$Q_2$$ are two fields that satisfy (1) and (2), then they are isomorphic. Such a field is called the field of fractions of $$D$$. This is the smallest field that contains a copy of $$D$$. Next we proved the field of fractions of a field $$F$$ is $$F$$. And we finished the lecture by proving that the field of fractions of the ring of Gaussian integers $$\mathbb{Z}[i]$$ is $$\mathbb{Q}[i]:=\{a+bi| a,b\in \mathbb{Q}\}$$.

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

We defined ideal; proved that $$I\unlhd \mathbb{Z}$$ if and only if $$I=n\mathbb{Z}$$ for some $$n$$; defined a finitely generated ideal, and proved $$\langle a_1,\ldots,a_n\rangle=\{\sum_{i=1}^n r_ia_i| r_i\in R\}$$ if $$R$$ is a unital commutative ring. Defined principal ideal and Principal Ideal Domain (PID). Indicated that $$\mathbb{Z}$$ is a PID. Proved that the kernel $$\ker f$$ of a ring homomorphism $$f:R\rightarrow S$$ is an ideal of $$R$$. Defined the factor ring of $$R$$ by $$I$$ when $$I\unlhd R$$.

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

We defined the factor ring $$R/I$$ of $$R$$ by its ideal $$I$$, and proved it is a ring; we defined the natural quotient map $$p:R\rightarrow R/I, p(r):=r+I$$, proved that $$p$$ is an onto ring homomorphism, and $${\rm ker}(p)=I$$; Proved The First Isomorphism Theorem: Suppose $$f:R\rightarrow S$$ is a ring homomorphism. Then (1) $${\rm ker}(f)$$ is an ideal of $$R$$; (2) $${\rm Im}(f)$$ is a subring of $$S$$; (3)$$\bar f: R/{\rm ker}(f)\rightarrow {\rm Im}(f), \bar f(r+{\rm ker})(f)):=f(r)$$ is an isomorphism. Using this we proved $$\mathbb{Z}/n\mathbb{Z}\simeq \mathbb{Z}_n$$ for any integer $$n\ge 2$$.

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

In this lecture, we discussed a couple of examples on how one can use the first isomorphism theorem. We emphasized on general techniques and the importance of having a generalized division algorithm for these methods to work. We proved that $$\mathbb{Q}[x]/\langle x^2+1\rangle\simeq \mathbb{Q}[i]$$ and have almost proved $$\mathbb{Z}[i]/\langle 3+2i\rangle\simeq \mathbb{Z}/13\mathbb{Z}.$$

• Lecture 11: Here is my note for the eleventh lecture.

In this lecture, first we finished proof of why $$\mathbb{Z}[i]/\langle 3+2i\rangle\simeq \mathbb{Z}/13\mathbb{Z}$$; these examples motivated us to define a Euclidean Domain. Then we proved $$\mathbb{Z}$$, $$F[x]$$ where $$F$$ is a field, and the ring $$\mathbb{Z}[i]$$ of Gaussian integers are Euclidean Domains. Next we stated that a Euclidean Domain is a PID, and pointed out the key idea on how to prove it. We will prove it next time.

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

In this lecture, first we proved a Euclidean Domain is a PID. Then we used this to prove $$\mathbb{Z}[i]/\langle a+bi\rangle \simeq \mathbb{Z}/p\mathbb{Z}$$ if $$a,b\in \mathbb{Z}$$ and $$a^2+b^2=p$$ is prime. Next we defined a prime ideal, and proved an ideal $$I$$ of $$\mathbb{Z}$$ is prime if and only if $$I$$ is either $$\{0\}$$ or $$p\mathbb{Z}$$ where $$p$$ is a prime number. We stated that for a unital commutative ring $$R$$ an ideal $$I$$ is prime if and only if $$R/I$$ is an integral domain. And we will prove this in the next lecture.

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

In this lecture, we proved that $$R/I$$ is an integral domain if and only if $$I$$ is a prime ideal; we defined maximal ideal, and proved that $$R/I$$ is a field if and only if $$I$$ is a maximal ideal; in particular a maximal ideal is a prime ideal. We showed an ideal $$I$$ of $$\mathbb{Z}$$ is maximal if and only if $$I=p\mathbb{Z}$$ where $$p$$ is a prime number. Next we defined an irreducible element, and we proved in a PID $$D$$ that is not a field $$\langle a\rangle$$ is a maximal ideal if and only if $$a$$ is an irreducible element.

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

In this lecture, first we saw how to use irreducible elements to show certain rings are not PID, for instance we showed $$\mathbb{Z}[\sqrt{-6}]$$ is not a PID; then we defined algebraic and transcendental numbers; proved that for any algebraic number $$\alpha\in \mathbb{C}$$ there is an irreducible polynomial $$m_{\alpha}(x)\in \mathbb{Q}[x]$$ such that $${\rm ker}(\phi_{\alpha})=\langle m_{\alpha}(x) \rangle$$, where $$\phi_{\alpha}:\mathbb{Q}[x]\rightarrow \mathbb{C}, \phi_{\alpha}(f(x)):=f(\alpha)$$ is the evaluation at $$\alpha$$ map; and so for $$f(x)\in \mathbb{Q}[x]$$ we have $$f(\alpha)=0$$ if and only if $$m_{\alpha}(x)|f(x)$$, and $$m_{\alpha}(x)$$ is a polynomial with smallest degree among the polynomials that have a zero at $$\alpha$$.

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

We proved the following main theorem:

Suppose $$\alpha\in \mathbb{C}$$ is an algebraic number and $$\phi_{\alpha}:\mathbb{Q}[x]\rightarrow \mathbb{C}, \phi_{\alpha}(f(x)):=f(\alpha)$$. Then

• (About the kernel) There is a unique monic irreducible polynomial $$m_{\alpha}(x)\in \mathbb{Q}[x]$$ such that $$\ker \phi_{\alpha}=\langle m_{\alpha}(x)\rangle$$; and if $$p(x)$$ is irreducible in $$\mathbb{Q}[x]$$ and $$p(\alpha)=0$$, then $$p(x)=cm_{\alpha}(x)$$ for some $$c\in \mathbb{Q}\setminus \{0\}$$.
• (About the image) $${\rm Im}(\phi_{\alpha})$$ is a field.
• (About the image) $${\rm Im}(\phi_{\alpha})=\{a_{d-1}\alpha^{d-1}+a_{d-2}\alpha^{d-2}+\cdots+a_0| a_i\in \mathbb{Q}\}$$, where $$d:=\deg m_{\alpha}(x)$$.
• (Isomorphism) $$\mathbb{Q}[x]/\langle m_{\alpha}(x)\rangle \simeq \{a_{d-1}\alpha^{d-1}+a_{d-2}\alpha^{d-2}+\cdots+a_0| a_i\in \mathbb{Q}\}$$.
Then we revisited the following example: $$\mathbb{Q}[x]/\langle x^2+1\rangle\simeq \mathbb{Q}[i]$$. Then we formally defined the degree of a polynomial over an arbitrary ring and proved that $$\deg(fg)=\deg f+\deg g$$ for $$f,g\in D[x]$$ if $$D$$ is an integral domain.

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

We proved that, if $$D$$ is an integral domain, then $$D[x]$$ is an integral domain, and $$U(D[x])=U(D)$$; we pointed out that $$1-2x$$ is a unit in $$\mathbb{Z}_{16}[x]$$; next we proved for any unital commutative ring $$R$$, $$f(x)\in R[x]$$, and $$c\in R$$, there is $$q(x)\in R[x]$$ such that $$f(x)=q(x)(x-c)+f(c)$$; in particular, $$f(c)=0$$ if and only if there is $$q(x)\in R[x]$$ such that $$f(x)=(x-c)q(x)$$ (The Factor Theorem); using the factor theorem, we proved that if $$F$$ is a field and $$f(x)\in F[x]$$, then $$f(x)$$ has at most $$deg f$$-many zeros in $$F$$; then we emphasized that though a polynomial can be viewed as a function, but they are not the same. For instance, we proved Fermat's little theorem, which asserts that $$x,x^p\in \mathbb{Z}_p[x]$$ give us the same functions from $$\mathbb{Z}_p$$ to $$\mathbb{Z}_p$$ if $$p$$ is prime; but of course they are different polynomials: one of them has degree 1 and the other one has degree $$p$$.

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

To appreciate the power of polynomials, we proved $$x^p-x=x(x-1)\cdots (x-p+1)$$ in $$\mathbb{Z}_p[x]$$ if $$p$$ is prime; and then deduced Wilson's theorem: $$(p-1)!\equiv \pmod{p}$$ if $$p$$ is prime; then we proved $$\binom{p-1}{i} \equiv (-1)^i \pmod{p}$$ if $$p$$ is prime. Next we proved, if $$F$$ is a field and $$f(x)\in F[x]$$ has positive degree, then $$f$$ is irreducible in $$F[x]$$ exactly when it cannot be written as a product of two polynomials od positive degree; we pointed out the subtle difference of being irreducible in $$\mathbb{Q}[x]$$ and $$\mathbb{Z}[x]$$; we will come back to this issue later; we observed that if $$f(x)\in F[x]$$ has degree at least 2 and $$f$$ has a zero in $$F$$, then $$f$$ is reducible in $$F[x]$$ (where $$F$$ is a field).

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

First we proved that if $$F$$ is a field, $$f\in F[x]$$ and $$2\le \deg f\le 3$$, then $$f$$ is irreducible in $$F[x]$$ exactly when $$f$$ does not have a zero in $$F$$; we showed why any degree polynomial is reducible in $$\mathbb{R}[x]$$; proved the rational root theorem: suppose $$f(x)=\sum_{i=0}^n a_ix^i\in \mathbb{Z}[x]$$, $$b,c\in \mathbb{Z}$$, $$\gcd(b,c)=1$$, and $$f(b/c)=0$$. Then $$b|a_0$$ and $$c|a_n$$; we mentioned a few corollaries of this result. We deduced: if a monic integer polynomial $$f$$ does not have a zero in $$\mathbb{Z}_p$$ for some prime $$p$$, then $$f$$ does not have a zero in $$\mathbb{Q}$$.

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

We used Fermat's little theorem to show certain large degree polynomials do not have a zero in $$\mathbb{Z}_p$$ for a small prime $$p$$, and deduced thar they do not have a zero in $$\mathbb{Q}$$; we formulated the mod $$p$$ irreducibility criterion, and to prove this we defined the content of an integer polynomial; and proved: for any $$f(x)\in \mathbb{Z}[x]\setminus \{0\}$$ there is a primitive polynomial $$\overline{f}(x)$$ such that $$f(x)=\alpha(f) \overline{f}(x)$$; $$p|\alpha(f)$$ if and only if $$c_p(f)=0$$; for a positive integer $$c$$, $$\alpha(cf)=c\alpha(f)$$.

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

We proved Gauss's lemma; using Gauss's lemma, we proved: suppose $$f(x)$$ is primitive and $$f(x)=\prod_{i=1}^m g_i(x)$$ for some $$g_i\in \mathbb{Q}[x]$$. Then there are $$a_i\in \mathbb{Q}\setminus \{0\}$$ such that (1) $$\prod_{i=1}^m a_i=1$$, (2) $$\overline{g}_i(x):=a_ig_i(x)$$ is primitive; in particular, $$f(x)=\prod_{i=1}^m \overline{g}_i(x)$$ and $$\deg \overline{g}_i=\deg g_i$$. Based on this result, we showed that a primitive polynomial is irreducible in $$\mathbb{Z}[x]$$ if and only if it is irreducible in $$\mathbb{Q}[x]$$. Next we started proof of the Mod $$p$$ Irreducibility Criterion.

• Lecture 21: Here is my note for the twenty first lecture.

• Lecture 22: Here is my note for the twenty second lecture.

• Lecture 23: Here is my note for the twenty third lecture.

• Lecture 24: Here is my note for the twenty fourth lecture.

• Lecture 25: Here is my note for the twenty fifth lecture.

• Lecture 26: Here is my note for the twenty sixth lecture.

• Lecture 27: Here is my note for the twenty seventh lecture.

Homework

• Homework are due on Wednesdays at 5:00 pm. You should drop your homework assignment in the homework drop-box in the basement of the AP&M building.
• 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:
• Your name (last name first).
• 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.
• Late Homework are not accepted.
• There will be 8 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: Wednesday, April 25, 12:00p-12:50p.
• Location: Students in sections 1 and 2 should go to Center Hall 216; and Students in sections 3 and 4 should go to Center Hall 212.
• Topics: All the topics that are covered till the end of lecture nine. You should know how to solve problems similar to the HW assignments 1,2, and 3.
• 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: Here is solution of the first exam.
• The second exam:
• Time: Wednesday, May 16, 12:00p-12:50p.
• Location: Students in sections 1 and 2 should go to Center Hall 216; and Students in sections 3 and 4 should go to Center Hall 212.
• Topics: All the topics that are covered till the end of lecture seventeen. You should know how to solve problems similar to the HW assignments 1, 2, 3, 4, 5, and 6.
• 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: Here is solution of the second exam.
• The final exam:
• Date: Wednesday, June 13, 11:30a- 2:29p.
• Location: Students in sections 1 and 2 should go to Center Hall 216; and Students in sections 3 and 4 should go to Center Hall 214.
• Topics: All the topics that were discussed in class, your homework assignments, and relevant examples and exercises in your book. (You will not be asked on the uniqueness of splitting fields).
• 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 help : TA's will have review sessions on Tuesday 9-11a and 1-3p.