Lectures:     M-W-F  12:00 PM--1:50 PM     Peter 103 Office Hour:     M-W-F  2:00 PM--3:30 PM   APM 7230

Discussion: TuTh 2:00 PM--2:50 PM  APM B412 TA: Sara Kumar (sak018ucsd edu) Office hour: Th 3:00 PM--5:00 PM 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. 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 6: 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 13: 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 20: 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 27: 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 3: 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.

Summary of my lectures You are supposed to read the relevant sections of your book. Here are parts of my lecture notes.

8/3. (1), Chapter 22 We pointed out that a large part of algebra has been about the study of zeros of polynomials. Then, we recalled the definition of a group, a cyclic group, and the order of an element. Proved that \({\rm ord}(g)=d\) if and only if \(g^n=e\) precisely when \(d|n\). Used this to show \(\langle g\rangle=\{e,g,\ldots,g^{{\rm ord}(g)}\}\), and \(g^{n_1}=g^{n_2}\) if and only if \(n_1\equiv n_2 \pmod{{\rm ord}(g)}\). Recalled basic properties of congruence arithmetic, and \(\mathbb{Z}_n\). Proved that a cyclic group of order \(n\) is isomorphic to \(\mathbb{Z}_n\). 8/5. (2), Chapter 22 and 23 Stated all the results needed from group theory: cosets, index of a subgroup, Lagrange's theorem, quotient of a group, kernel and image of a group homomorphism, and the fundamental theorem of group Homomorphisms. Defined rings. Saw some exmples and non-examples! Defined unital and commutative rings. Mentioned some examples. Defined the matrix ring over a ring \(R\). The ring of polynomials over a ring \(R\). Proved that in a unital ring identity is unique. Proved that in every ring \(0\cdot a=a\cdot 0=0\), \((-a)\cdot b=a\cdot (-b)=-(a\cdot b)\), and \((-a)\cdot(-b)=a\cdot b\). Units of a ring. Find units of the Gaussian integers. Defined ring Homomorphism and proved that \(c\colon \mathbb{Z}\to R,\quad c(n):=n1_R\) is a ring homomorphism. Defined zero-divisors. 8/7. (3), Chapter 22 and 23 Defined the characteristic of a ring. Proved that if the lcm of order of elements of \(A\) is a positive integer \(n\), then the characteristic of \(A\) is \(n\). If the lcm of the orders of elements of \(A\) is infinite, then the characteristic of \(A\) is zero. For a unital ring, we proved that the characteristic of \(A\) is the order of \(1_A\) given it is finite. If the order of \(1_A\) is infinite, then \(A\) is of characteristic zero. Mentioned a few examples: \(\mathbb{Z}\) is of characteristic zero. \(\mathbb{Z}_n\) is of characteristic \(n\). \(m\mathbb{Z}_n\) is of characteristic \(n/\gcd(n,m)\). We defined what a ring homomorphism is. Proved that if \(A\) is a unital ring, then \(c:\mathbb{Z}\to A, \quad c(n):=n 1_A\) is a ring homomorphism. Pointed out that a ring homomorphism is also an additive group homomorphism. Recalled what the kernel of a group homomorphism is. we considered two positive integers \(m, n\) and showed that the map \(c_{n,m} : \mathbb{Z}_n \to \mathbb{Z}_m, \quad c_{n,m}([a]_n) = [a]_m\) is a well-defined ring homomorphism if and only if \(m \mid n\). We used this result to prove the Chinese Remainder Theorem (CRT): if \(\gcd(m, n) = 1\), then the map \(\mathbb{Z}_{mn} \to \mathbb{Z}_m \times \mathbb{Z}_n, \quad [a]_{mn} \mapsto ([a]_m, [a]_n)\) is a ring isomorphism. Showed that \(\mathbb{Z}_n^\times = \{ [a]_n \mid \gcd(a, n) = 1 \}.\) This led us to define Euler’s \(\phi\)-function. Proved that \(\phi(mn) = \phi(m)\phi(n)\) whenever \(\gcd(m, n) = 1\) and \(\phi(p)=p-1\) if \(p\) is prime. Defined a field and proved that \(\mathbb{Z}_n\) is a field if and only if \(n\) is prime. 8/10. (4), Chapter 23 and 26 Recalled\(\phi(mn)=\phi(m)\phi(n)\) if \(\gcd(m,n)=1\). Proved \(\phi(p^k)=p^k-p^{k-1}\) if \(p\) is prime. Mentioned how to use these to find \(\phi(n)\) for an arbitrary \(n\). Defined integral domains. Proved that a non-trivial unital commutative ring is an integral domain if and only if it satisfies the cancellation law. Proved that a field is an integral domain. Pointed out the converse does not always hold. But a finite ring is an integral domain if and only if it is a field. Next, we showed that the characteristic of an integral domain is either zero or prime. Mentioned that \(\mathbb{Z}_n\) is an integral domain if and only if \(n\) is prime if and only if \(\mathbb{Z}_n\) is a field. Defined the field of fractions of an integral domain. 8/12. (5), Chapter 26 Defined the field of fractions of an integral domain, and formulated and proved its universal property. We had our first exam. 8/14. (6), Chapter 26 and 27 Gave a pseudo algorithm to prove that \(Q(D)\simeq F\). Applied this method to show that \(Q(\mathbb{Z}[i])\simeq \mathbb{Q}[i]\). Defined the degree and the leading coefficient of a polynomial. Proved that over an integral domain, \(L(fg)=L(f)L(g)\), \(\ell(fg)=\ell(f)\ell(g)\), and \(\deg(fg)=\deg f+\deg g\). Used this to show that if \(D\) is an integral domain, then (1) \(D[x]\) is an integral domain, and (2) \(D[x]^\times=D^\times\). Next, we defined a nilpotent element. Proved that if \(a\) is nilpotent, then \(1\pm a\) is a unit. Used this to show that \(1+2x\) is a unit in \(\mathbb{Z}_{8}[x]\) and find its inverse. Mentioned that in general, the group of units of \(A[x]\) consists of polynomials of the form \(a_0+a_1x+\cdots+a_nx^n\) such that \(a_0\in A^\times\) and \(a_1,\ldots,a_n\) are nilpotents. Stated the long division algorithm over an arbitrary unital commutative ring: suppose \(f,g\in R[x]\) and \(\ell(g)\in R^\times\). Then there exists a unique pair \((q,r)\) such that \( f(x) = q(x)g(x) + r(x) \) and \( \deg r < \deg g \). Then, we proved the existence part. 8/17. (7), Chapter 27 and 28 Division algorithm, factor theorem, generalized factor theorem, number of distinct zeros over an integral domain. 8/19. (8), Chapter 28 Zeros of polynomials in direct products, Fermat's little theorem, Frobenius homomorphism, factorization of \(x^p-x\) over \(\mathbb{Z}_p\), Wilson's theorem, irreducible elements, irreducible polynomials over a field, having zero and factorization, degree \(2\) or \(3\) irreducibility criterion. 8/21. (9), Chapter 28 Rational root criterion and its applications. No zero mod \(p\) implies no rational zero. Content of a polynomial and its basic properties. 8/24. (10), Chapter 28 and 30 Gauss's lemma, versions 1,2, and 3. Mod-\(p\) irreducibility criterion. Eisenstein's irreducibility criterion. 8/26. (11), Chapter 30 and 31 Ideal, ideals generated by a given set of elements, principal ideals, kernels are ideals, ideals of a field, PID. second exam. 8/28. (12), Chapter 30 and 31 quotient ring, fundamental theorem of homomorphism, evaluation map revisited, algebraic elements. 8/31. (13), Chapter 31 Minimal polynomial, correspondence theorem, maximal ideals, irreducible elements in a PID and maximal ideals 9/2. (14), Chapter 31 Maximal ideals of PIDs and irreducible elements. Image of the evaluation map at an algebraic element. Prime ideals. Prime elements. Connection between prime and irreducible 9/4. (15), Unique Factorization Domains and Finite fields

My old notes. 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 12, 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 22 and 23 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. Practice: In addition to reviewing your homework assignments, in-class examples, and the relevant textbook chapters, you may find the following practice exams helpful for further preparation: Summer 2017, PDF . Winter 2019, PDF . Exam: The second exam: Time: Wednesday, August 26, 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 22, 23, 26, 27, 28, and 30. 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 30) 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. Practice: In addition to reviewing your homework assignments, in-class examples, and the relevant textbook chapters, you may find the following practice exams helpful for further preparation: Summer 2017, PDF . Winter 2019, PDF . Exam: The final exam: Date: Saturday, September 5, 3:00 PM--5:59 PM Location: the usual classroom. 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. Summary of lectures You may find this summary useful. Practice: In addition to reviewing your homework assignments, in-class examples, and the relevant textbook chapters, you may find the following practice exams helpful for further preparation: Summer 2017, PDF . Winter 2019, PDF .