Modern Algebra II (Math 103 B)

Summer 2017

 Lectures: M-W-F 9:00 AM--10:50 AM PCYNH 122 Office Hour: M-W-F 11:00 PM--12:00 PM APM 7230
 Discussion: TuTh 8:00 AM--8:50 AM APM B402A TA: Artem Mavrin (amavrinucsd edu) Office hour: Day Time Location
 Discussion: TuTh 9:00 AM--9:50 AM APM B402A TA: Artem Mavrin (amavrinucsd edu) Office hour: Day Time Location
 Discussion: TuTh 10:00 AM--10:50 AM APM B402A TA: Zach Higgins (zhigginsucsd edu) Office hour: TuTh 11:00 AM--12:00 PM APM 5829

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

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. I would like to thank Zach Higgins for providing the solutions to the homework sets. (Solutions will not be necessarily available after the course is finished.)

• Due August 10: Here is the first problem set. Here are the solutions to 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 August 17: Here is the second problem set. Here are the solutions to 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 August 24: Here is the third problem set. Here are the solutions to the third problem set. (Special thanks to Zach, who had been working on these solutions at 2:00 am to make sure that you have something that can help you for the exam.)

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 August 31: Here is the fourth problem set. Here are the solutions to the fourth 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 23, problems 12, 14, 16, 17, 18, 19, 21, 22, 30, 31.
• Section 26, probelms 1, 2, 3, 17, 18, 26, 37.
• Section 27, problems 31.

• Due September 7: Here is the fifth problem set. Here are the solutions to the fifth 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 26, probelm 38.
• Section 27, problems 1, 2, 3, 4, 5, 6, 8, 15, 16, 17, 18, 19, 24, 30.
• Section 29, problems 1, 2, 3, 4, 6, 8, 18, 25, 30, 32.

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, and commutative and unital rings. Congruence arithmetic is recalled. 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.

• Here is the lecture note for the second lecture.

In this lecture, we showed some basic properties of operations in any ring; defined the characteristic of a unital ring; showed that for any unital ring $$R$$ there is a canonical homomorphism $$c:\mathbb{Z}\rightarrow R$$; discussed homomorphisms between $$\mathbb{Z}_n$$'s; proved Chinese Remainder Theorem; defined the Euler $$\phi$$-function and proved it is multiplicative;

• Here is the lecture note for the third lecture.

In this lecture, we defined division ring and field; defined integral domain; proved that a finite integral domain is a field; showed that $$\mathbb{Z}_n$$ is an integral domain if and only if $$n$$ is prime; defined the field of fractions (we will continue its proof in the next lecture).

• Here is the lecture note for the fourth lecture.

In this lecture, we defined the field of fractions of an integral domain, and presented a few examples on this. The ring of polynomials with coefficients in a given ring is also defined.

• Here is the lecture note for the fifth lecture.

In this lecture, we defined degree of a polynomial; showed that $$\deg(fg)=\deg(f)+\deg(g)$$ for polynomials over an integral domain; described the group of units of a ring of polynomial. We compared polynomials vs functions; proved Fermat's theorem; and introduced the evaluation homomorphism.

• Here is the lecture note for the sixth lecture.

In this lecture, we studied further the evaluation homomorphisms and their connection with zeros of polynomials; proved the division algorithm; factor theorem; and the fact that a polynomial $$f$$ with coefficients in an integral domain has at most $$\deg(f)$$ zeros.

• Here is the lecture note for the seventh lecture.

In this lecture, we said what an irreducible element of a unital commutative ring is; described irreducible elements of $$F[x]$$ where $$F$$ is a field; showed that a degree 2 or 3 polynomial in $$F[x]$$ is reducible exactly when it has a zero in $$F$$; described a method to effectively find possible rational zeros of a polynomial with integer coefficients; extended the residue maps to the ring of polynomials $$\mathbb{Z}[x]$$; showed the following: Suppose $$f(x)$$ in $$\mathbb{Z}[x]$$ has degree at least 2 and its leading coefficient is 1. If $$c_p(f)$$ is irreducible in $$\mathbb{Z}_p[x]$$ (for some prime $$p$$), then $$f$$ is irreducible in $$\mathbb{Q}[x]$$; we showed how Fermat's theorem can be used to find possible zeros of a polynomial with large degrees in $$\mathbb{Z}_p$$.

• Here is the lecture note for the eighth lecture.

In this lecture, we proved Gauss's lemma and then used it to prove two irreducibility criteria; one of them is known as Eisenstein Criterion. We also defined what an ideal is.

• Here is the lecture note for the ninth lecture.

In this lecture, we showed that a proper ideal does not have a unit; only ideals of a field $$F$$ are 0 and $$F$$; ideals of $$\mathbb{Z}$$ are of the form $$n\mathbb{Z}$$ for some intgere $$n$$; defined what a Principal Ideal Domain is; proved that the ring of polynomials $$F[x]$$ with coefficients in a field $$F$$ is a PID; proved that the kernel of a ring homomorphism is an ideal; defined the quotient ring and the natural quotient map; deduced that a subset of a ring $$R$$ is an ideal exactly when it is kernel of a ring homomorphism of $$R$$.

• Here is the lecture note for the tenth lecture.

In this lecture, we proved the Fundamental Homomorphism Theorem (also known as the First Isomorphism Theorem); proved $$\mathbb{Z}/n\mathbb{Z}$$ is isomorphic to $$\mathbb{Z}_n$$ as a ring; studied the evaluation map at an algebraic number via an example: showed that the kernel of $$\phi_{\sqrt{2}}$$ is generated by $$x^2-2$$; the image of $$\phi_{\sqrt{2}}=\mathbb{Q}[\sqrt{2}]$$; and $$\mathbb{Q}[x]/\langle x^2-2\rangle\simeq \mathbb{Q}[\sqrt{2}]$$; proved that if $$\alpha\in \mathbb{C}$$ is an algebraic number, then there is an irreducible polynomial $$m_{\alpha}(x)\in \mathbb{Q}[x]$$ which generates the kernel of the evaluation map $$\phi_{\alpha}:\mathbb{Q}[x]\rightarrow \mathbb{C}$$ at $$\alpha$$.

• Here is the lecture note for the eleventh lecture.

In this lecture, we proved that the image of the evaluation map $$\phi_{\alpha}$$ at an algebraic number $$\alpha$$ which is a zero of a polynomial of degree $$d_0$$ is $$\mathbb{Q}[\alpha]:=\{c_0+c_1 \alpha+\cdots+ c_{d_0-1} \alpha^{d_0-1}|\hspace{1mm} c_i\in \mathbb{Q}\}$$; To prove $$\mathbb{Q}[\alpha]$$ is a field, we investigated the ideals $$I$$ of a unital commutative ring $$R$$ with the property that $$R/I$$ is a field; and we proved that $$R/I$$ is a field if and only if $$I$$ is a maximal ideal of $$R$$ (here $$R$$ is a unital commutative ring.).

• Here is the lecture note for the twelfth lecture.

In this lecture, we proved if $$D$$ is a PID and $$a\in D\setminus \{0\}$$, then $$\langle a\rangle$$ is maximal if and only if $$a$$ is irreducible in $$D$$. Using this result we finished the proof of the following theorem.

Suppose $$\alpha\in \mathbb{C}$$ is an algebraic number. Then there is a polynomial $$m_{\alpha}(x)\in \mathbb{Q}[x]$$ such that (1) $$m_{\alpha}(x)$$ is irreducible in $$\mathbb{Q}[x]$$, (2) for $$f(x)\in \mathbb{Q}[x]$$, if $$\alpha$$ is a zero of $$f(x)$$, then $$m_{\alpha}(x)$$ divides $$f(x)$$; that means $$f(x)=m_{\alpha}(x)q(x)$$ for some $$q(x)\in \mathbb{Q}[x]$$. And $$\mathbb{Q}[\alpha]:=\{c_0+c_1\alpha+\cdots+c_{d-1}\alpha^{d-1}|c_i\in \mathbb{Q}\}$$ is a field where $$d=\deg m_{\alpha}$$.

Next we defined what a prime ideal is; and showed that in a unital commutative ring $$R$$ an ideal $$I$$ is prime if and only if the quotient ring $$R/I$$ is an integral domain. In the rest of the lecture we proved the following theorem.

Let $$F$$ be a field and $$p(x)$$ be an irreducible polynomial in $$F[x]$$. Then there are a field $$E$$, an embedding $$i:F\rightarrow E$$, and $$\alpha\in E$$ such that $$\alpha$$ is a zero of $$i(p)(x)$$ (which essentially means $$p(x)$$, but viewed as an element of $$E[x]$$ via the embedding $$i$$.)

• Here is the lecture note for the thirteenth lecture.

In this lecture, we defined what a Unique Factorization Domain is, and proved that a PID is a UFD; the existence part is only proved for $$F[x]$$, but you can find the argument for the general case in the lecture note. Along the way certain properties of irreducible elements of a PID is also proved.

• Here is the lecture note for the fourteenth lecture.

In this lecture, we proved that, if $$f(x)\in \mathbb{Z}_p[x]$$ is an irreducible polynomial of degree $$d_0$$, then there are a field extension $$E$$ of $$\mathbb{Z}_p$$ and $$\alpha\in E$$ such that $$\alpha$$ is a zero of $$f(x)$$; moreover $$\mathbb{Z}_p[\alpha]=\{c_0+c_1\alpha+\cdots+c_{d_0-1}\alpha^{d_0-1}|\hspace{1mm} c_i\in \mathbb{Z}_p\}$$ is a field and $$|\mathbb{Z}_p[\alpha]|=p^{d_0}$$. Hence if there is an irreducible polynomial of degree $$d_0$$ in $$\mathbb{Z}_p[x]$$, then there is a finite field of order $$p^{d_0}$$. In the rest of the lecture, we went through some of the old topics.

Homework

• Homework are due on Thursdays at 3:00 pm. You should drop your homework assignment in the homework drop-box in the basement of the AP&M building.
• Late Homework are not accepted.
• There will be 5 problem sets. Your cumulative homework grade will be based on the best 4 of the 5.
• 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.
• You must bring a blue book to the exams.
Exams.

• The first exam:
• Time: Wednesday, August 16, 10:00-11:50.
• Location: this is an in-class exam. (the second half of the lecture).
• Topics: All the topics that are discussed in class and Sections 18, 19, and 21 from the main book of the course.
• 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 the first exam.
• Will be posted here.
• The second exam:
• Time: Wednesday, August 30, 10:00-11:50.
• Location: this is an in-class exam. (the second half of the lecture).
• 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 $$x^p=x$$ for any $$x\in \mathbb{Z}_p$$ if $$p$$ is prime, and $$F[x]$$ is a PID when $$F$$ is a field are not covered in the mentioned Sections, but they might be part of the exam. But the Fundamental Homomorphism Theorem (which is discussed in Section 26) is not part of this exam. Any topic discussed in the mentioned Sections except the Fundamental Homomorphism Theorem might be part of the exam.
• 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 the second exam.