Lectures:     M-W-F  9:00 AM--10:50 AM and 11:00 AM--12:50 PM    Peter 102 and APM 5402 Office Hour:     M-W-F  1:00 PM--2:30 PM   APM 7230

Discussion: TuTh 9:00 AM--9:50 AM  PODEM 1A23 TA: Hugo Jenkins (hajenkinsucsd edu) Office hour: TuTh 1-2 pm HSS 5005

Discussion: TuTh 10:00 AM--10:50 AM  Centr 218 TA: Toren D.Nelly-Warady (tdnellywaradyucsd edu) Office hour: Wednesdays 2-4 HSS 4012

Discussion: TuTh 12:00 AM--12:50 AM  APM 5402 TA: Hargun Bhatia (hpbhatiaucsd edu) Office hour: Thursdays 2-4pm HSS 5012

Book John B. Fraleigh, A first course in Abstract Algebra 8th 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. Due August 8: 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 22, problems 1, 3, 6, 8, 19, 20, 21, 22, 23, 24, 25, 27, 28, 32, 38, 40, 44. Due August 15: 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 23, problems 2, 3, 10, 12, 14, 30. Section 24, problem 27. Due August 22: 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 26, problems 1, 2, 17. Section 27, problems 4, 7, 11, 16, 17, 25, 27. Section 28, problems 1, 2, 3, 4, 6, 8, 9, 10, 11, 26, 34, 35, 36. Due August 29: Here is 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 28, problems 12, 14, 16, 17, 18, 19, 21, 22, 30, 31. Section 30, probelms 1, 2, 3, 17, 18, 26, 37. Due September 5: Here is 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 31, problems 1, 2, 3, 4, 5, 6, 8, 15, 16, 17, 18, 19, 24, 30. Section 34, problems 1, 2, 3, 4, 6, 8, 25, 30. Section 39, problems 25, 30, 34.

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 9:00 pm. You should post them in GradeScope. Late Homework are not accepted. Selected problems on the each assignment will be graded. Style: A messy and disorganized homework might get no points. You can scan or simply take a clear photo of your homework and upload it. You must select pages corresponding to your solutions of problems during the upload process. If you have not selected pages when the grader begins grading, the grader will not grade your assignment and you will receive a grade of 0 on it. No appeals of this policy will be considered. 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. You can use Overleaf. Overleaf is an easy to use and an excellent online LaTeX editor. 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. 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.

Grade 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+.

Further information There is no make-up exam. No notes, textbooks, calculators and electronic devices are allowed during exams.

Exams. The first exam: Time: Wednesday, August 14, during the lecture. 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: The second exam: Time: Wednesday, August 28, during the lecture. 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: The final exam: Date: Saturday, September 7, 8:00-10:59 and 11:30-14:29 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.