Math 100C: Algebra III

Spring 2021

Lectures: Will be pre-recorded and posted in YouTube.
The scheduled lecture time will be used as a Discussion and Problem session.

Discussion and Problem sessions:
T, Th   12:30p-1:50p
Meeting ID:   Password:
985 8662 0622   The cardinality of the symmetric group \(S_7\) times the cardinality of \({\rm Aut}_{\mathbb{F}_2}(\mathbb{F}_{64})\).
Office hours:
T, Th               1:50-2:15
TA's information:
Name   E-mail adddress   Meeting ID
Alexander Mathers   ucsd edu   991 9495 4695
TA's office hours: Your scheduled discussion session or Friday 4-5pm.

General information     Book     Calendar     Lecture     Homework     Quizzes     Grade     Regrade     Assignment
General information

  • Title: Abstract Algebra III: Introduction to Galois Theory and commutative algebra.
  • Credit Hours: 4.
  • Prerequisite: Math 100b. Math 100 is a difficult and time consuming course, so enroll only if your course load allows it.
  • Catalog Description: Third course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. In this course, we study basics of field theory, module theory, and commutative algebra from the point of view of understanding zeros of polynomials.

  • Michael Artin, Algebra.
  • There are lots of interesting books on ring theory. I like the following books:
    • J. J. Rotman, A first course in abstract algebra.
    • D. S. Dummit, R. M. Foote, Abstract algebra. More advanced.
    • T. W. Hungerford, Algebra. More advanced.
    • M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra. More advance.

We will continue with the lecture notes of math 100 b and we add to it.

Here is the lecture notes for this course. You are expected to study the lecture notes and watch the posted lectures prior to the Discussion and Problem sessions.

Here is the topics for the Discussion and Problem Sessions for this course. The Zoom videos are shared via Canvas.

Lectures and Topics by date

  • T 3/30 Some results from field theory are reviewed. Group action, orbits, stabilizer subgroup, the orbit-stabilizer theorem were recalled. It is proved that a finite field extension \(E/F\) is a Galois extension if and only if for every \(\alpha\in E\) the \({\rm Aut}_F(E)\)-orbit of \(\alpha\) has exactly \(\deg m_{\alpha,F}\)-many elements. To find a connection between subgroups of the group of symmetires of a finite extension and intermediate subfields, we showed that for every finite subgroup \(G\) of \({\rm Aut}_F(E)\), \(E/{\rm Fix}(G)\) is a Galois extension and for every \(\alpha\in E\), \(m_{\alpha,{\rm Fix}(G)}(x)=\prod_{\alpha'\in G\cdot \alpha} (x-\alpha')\).
    Here is the link to the 1st lecture.
  • Th 4/1 Suppose \(E\) is a field and \(G\) is a finite subgroup of \({\rm Aut}(E)\). Then \({\rm Fix}(G)/F\) is a Galois extension of index \(|G|\) and \({\rm Aut}_{{\rm Fix}(G)}(E)=G\). Fundamental Theorem of Galois theory is proved. We showed that if \(E/F\) is a finite separable extension, then \(E/F\) has only finitely many intermediate subfields.
    Here is the link to the 2nd lecture.
  • T 4/6 Fundamental theorem of algebra is proved. For a finite extension \(E/F\), we have that \({\rm Int}(E/F)\) is finite if and only if \(E/F\) is a simple extension. If \(E/F\) is a finite separable extension, then \(E/F\) is a simple extension. For an algebraic extension \(E/F\), we defined the separable extension of \(F\) in \(E\) and denoted this by \(E_{\rm sep}\). We proved that \(E_{\rm sep}/F\) is a separable extension, \(E=E_{\rm sep}\) if the characteristic of \(F\) is zero, and \(E^\times/E_{\rm sep}^\times\) is a \(p\)-group if the characteristic of \(F\) is \(p>0\).
    Here is the link to the 3rd lecture.
  • Th 4/8 Purely inseparable extensions are introduced and their main properties are proved. We prove that for every algebraic extension \(E/F\), \(E/E_{\rm sep}\) is purely inseparable and \(E_{\rm sep}/F\) is separable. For an algebraic extension \(E/F\) and an intermediate subfield \(K\) of \(E/F\), we prove that \(E/F\) is separable if and only if \(E/K\) and \(K/F\) are separable. Polynomials that are solvable by radicals are defined. Motivated by that we also defined radical extensions. Looking at the building blocks of radical extensions, we recalled Kummer extensions. To get the Kummer extensions, we need to have enough roots of unity. To that end, we recalled generalized cyclotomic extensions.
    Here is the link to the 4th lecture.
  • T 4/13 If \(L/E\) and \(E/F\) are radical extensions, then \(L/F\) is a radical extension. If \(L\) is a splitting field of \((x^n-a_1)\cdots (x^n-a_m)\) for some \(a_i\)'s in \(F\), then \(L/F\) is a Galois radical extension. If \(E/F\) is a normal extension, \(f\in F[x]\), and \(L\) is a splitting field of \(f\) over \(E\), then \(L/F\) is a normal extension. If \(E/F\) is a radical extension and characteristic of \(F\) is zero, then there is \(L/E\) such that \(L/F\) is a Galois radical extension. (Galois) If \(F\) is a field of characteristic zero and \(f\in F[x]\) is solvable by radicals over \(F\), then \({\rm Aut}_F(K)\) is a solvable group where \(K\) is a splitting field of \(f\) over \(F\).
    Here is the link to the 5th lecture.
  • Th 4/15 \(G\) is solvable if and only if \(G/N\) and \(N\) solvable. If \(G\) is solvable, then every subgroup of \(G\) is solvable. A solvable simple group is a cyclic group of prime order. \(A_n\) and \(S_n\) are no solvable if \(n\geq 5\). If \(f\in \mathbb{Q}[x]\) has a prime degree \(p\), is irreducible, has \(p-2\) real and \(2\) non-real complex zeros, then the Galois group of \(f\) over \(\mathbb{Q}\) is isomorphic to \(S_p\). There is a degree 5 polynomial which is not solvable by radicals over \(\mathbb{Q}\). If \(G\) is a finite solvable group, then there is a chain of subgroups \(G=:G_0\unrhd G_1 \unrhd \cdots \unrhd G_m=1\) such that \(G_i/G_{i+1}\)'s are cyclic groups of prime order.
    Here is the link to the 6th lecture.
  • T 4/20 We proved Dirichlet's independence of characters, Hilbert's Theorem 90, and Galois's solvability theorem. Along the way we showed that if \(E/F\) is a cyclic Galois extension of degree \(n\) and \(F\) is a characteristic zero field which contains an element of multiplicative order \(n\), then \(E/F\) is a Kummer extension; that means \(E=F[\sqrt[n]{a}]\) for some \(a\in F\).
    Here is the link to the 7th lecture.
  • Th 4/22 In this shorter video, we discussed algebraically closed fields, axiom of choice, Zorn's lemma, and proved that every proper ideal of a unital ring is contained in a maximal ideal. This result was used to show that for every field \(F\) there is an algebraic extension \(K/F\) such that every non-constant polynomial \(f\in F[x]\) has a zero in \(K\). This result will be used to prove the existence of an algebraic closure of a field.
    Here is the link to the 8th lecture.
  • T 4/27 In this lecture, we finished the proof of existence of algebraic closure of a field, and proved the uniquness of algebraic closures. Basic properties of algebraic closures are proved as well.
    Here is the link to the 9th lecture.
  • Th 4/29 In this lecture, we proved a Galois correspodence theorem for algebraic closures. We defined Kummer pairing and started the proof of the cyclic case of Kummer's theorem.
    Here is the link to the 10th lecture.
  • T 5/4 In this lecture, we finished proof of the cyclic case of Kummer theory. We stated the finite abelian case of Kummer theory. Next we defined dual of a finite abelian group, and proved that the intersection of all the kernels of characters of a finite abelian group is tirivial; we showed this result without using classification of finite abelian groups.
    Here is the link to the 11th lecture.
  • Th 5/6 This is our last lecture on field theory. In this lecture which is recorded in two (short) videos, we prove that every finite abelian group has the same number of elements as its dual and every finite abelian group is naturally isomorphic to its dual of dual. Next we prove that the Kummer pairing is a pefect pairing. This part is the most difficult part of the proof. In the second video, we prove an important property of perfect pairing of finite abelian groups: being a perfect pairing can be formulated with respect to either components! We use these results to prove the finite abelian case of Kummer extensions.
    Here and Here are the links to the 12th lecture.
  • T 5/11 Catch up with the longer videos from previous lectures.
  • Th 5/13 In this lecture, motivated by the study of zeros of systems of linear equations over a ring, we define modules. Quotient of modules are defined and the first isomorphism theorem for modules is proved. Noetherian modules are defined and it is proved that all submodules of a Noetherian module are finitely generated.
    Here is the link to the 13th lecture.
  • T 5/18 In this lecture, we completed our proof that a module is Noetherian if and only if all of its submodules are finitely generated. Then we used this to show that \(A\) is Noetherian exactly when \(B\) and \(A/B\) are Noetherian for a submodule \(B\) of \(A\). From this we conclude that \(A+B\) is Noetherian if and only if submodules \(A\) and \(B\) are Noetherian. Based on these results, we prove that for a Noetherian ring \(R\) every finitely generated \(R\)-module is Noetherian. Using this result we prove that for a Noetherian ring \(R\), every finitely generated \(R\)-module is isomorphic to the co-image of a matrix. This takes us to system of linear equations over rings. We showed how we can use the row and the column reduction operations over Euclidean domains to get a Smith normal form of a matrix.
    Here is the link to the 14th lecture.
  • Th 5/20 In this lecture, using the existence of a Smith normal form over a Euclidean domain, we prove a classification of finitely generated modules over a Euclidean domain. As a special case, we obtain a classification of finitely generated and finite abelian groups. In the second half, we defined a matrix representation of a linear map, and formulated a question: can we find a sparse matrix representation of a linear map? We start with the evaluation map from the ring of polynomials to the ring of linear maps. Similar to the study of algebraic numbers, the minimal polynomial of a linear transformation is defined. We also show that two different matrix representations of a given linear map are conjugate of each other. We define minimal polynomial of a square matrix and show that the minimal polynomial does not change after conjugation. Finally we say how having a linear map \(T\) from \(V\) to \(V\), we can view \(V\) as a \(F[x]\)-module. This will be used later.
    Here is the link to the 15th lecture.
  • T 5/25 In this lecture, using classification of finitely generated modules over a Euclidean domain, we prove the existence of a rational canonical form, prove the Calyey-Hamilton theorem, and show that if \(p\in F[x]\) is irreducible, then \(p\) divides the minimal polynomial \(m_{A,F}(x)\) of a matrix \(A\) with entries in \(F\) if and only if \(p\) divides the characteristic polynomial \(f_A\) of \(A\).
    Here is the link to the 16th lecture.
  • Th 5/27 In this lecture, various forms of generalized long division are discussed and Hilbert's basis theorem is proved. Using Hilbert's basis theorem, we showed that every finitely generated ring and every finitely generated \(F\)-algebra, where \(F\) is a field, is Noetherian.
    Here is the link to the 17th lecture.
  • T 6/1 In this lecture, we start proving Hilbert's Nullstellensatz. Along the way the resultant of two polynomials in \(A[x]\) is defined and it is proved that for \(f,g\in A[x]\), \(r(f,g)\in \langle f, g \rangle\cap A\).
    Here is the link to the 18th lecture.
  • Th 6/3 In this lecture, various versions of Hilbert's Nullstellensatz are proved. As an application, we show that every prime ideal of the ring of multivariable polynomials with coefficients in an algebraically closed field is the intersection of its maximal divisors.
    Here is the link to the 19th lecture.


  • Homework will be assigned in the assignment section of this page.
  • Homework are due on Sundays at 9:00 pm except the last one, through GradeScope. The last homework is due on the last Friday of the quarter.
  • Late homework is not accepted.
  • There will be 10 problem sets. Your cumulative homework grade will be based on the best 9 of the 10.
  • 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.
  • A good portion of the exams will be based on the weekly problem sets and problems discussed in the D and P sessions. So it is extremely important for you to make sure that you understand each one of them.
  • 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.
  • Reading the sections of the lecture note corresponding to the assigned homework exercises is considered part of the homework assignment.

  • There will be 5 quizzes throughout the quarter.
  • You will write them on Thursdays 12:30-1:20 or 19-19:50. Before the first quiz, you have to let me know if you will be taking the quizzes at 12:30 or 19.
  • No collaboration with other humans or with online resources is allowed.
  • No (e-)notes, textbooks, and calculators are allowed during exams.
  • Questions are fairly similar to the homework assignments, the examples discussed in the class, and problems discussed in the D and P sessions. Make sure that you know how to solve anyone of them.
  • Solutions should be clearly written and you must select pages corresponding to your solutions of problems during the upload process. For the quizzes, your solutions should be hand-written. It can be done on your tablets.
  • The link to quizzes will be given during the Zoom. Students will be divided in two main groups, and will be assigned different exams accordingly.
  • Students who take the earlier quiz are not allowed to share their exams or discuss the problems with others till the second group is done with the quiz.
  • You should use the same Zoom link as the one provided for the Discussion And Problem Sessions. Either your TA or I will be on the Zoom meeting. Your cameras should be on for the duration of quizzes. Students with the AFA will be assigned to separate breakout rooms.

  • The first quiz:
    • Date: 4/8
    • Topics: All the topics that are covered in the first 2 lectures.
    • Here is the 1st quiz.
    • Here are (outline of) solutions which are provided by Alex.
  • The second quiz:
    • Date: 4/22
    • Topics: All the topics that are covered in the first 6 lectures.
    • Here is the 2nd quiz.
    • Here are (outline of) solutions which are provided by Alex.
  • The third quiz:
    • Date: 5/6
    • Topics: All the topics that are covered in the first 10 lectures.
    • Here is the 3rd quiz.
    • Here are (outline of) solutions which are provided by Alex.
  • The fourth quiz:
    • Date: 5/20
    • Topics: All the topics that are covered in the first 12 lectures..
    • Here is the 4th quiz.
    • Here are (outline of) solutions which are provided by Alex.
  • The fifth quiz:
    • Date: 6/7 (Notice that this is the Monday of the final exam week; times are still 12:30-1:20 or 19-19:50.)
    • Topics: All the topics that are covered.

  • There will be 5 quizzes. Your cumulative quiz grade will be based on the best 4 of the 5.
  • Your final weighted score is
    • Homework 20%+ Quiz grade 80%
  • 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-
      97 93 90 87 83 80 77 73 70
    •  Based on a curve where the median corresponds to the cut-off B-/C+.
  • If more than 90% of the students fill out the CAPE questioner at the end of the quarter, all the students get one additional point towards their weighted score.
  • If you wish to have your homework or quizzes regraded, you must request regarde through Gradescope within the specified window of time. No regrade will be accepted after the deadline.
  • Please do not enter an erroneous regrade request on Gradescope, i.e. do not ask for a regrade without a good reason (and please explain your reasoning in your request).
  • Submitting a regrade request without a legitimate explanation may result in the loss of one point on the given problem.
Further information
  • There is no make-up exam.
  • No notes, textbooks, calculators and electronic devices are allowed during exams.
  • Academic Integrity: Academic dishonesty is considered a serious offense at UCSD. Students caught cheating will face an administrative sanction which may include suspension or expulsion from the university. It is in your best interest to maintain your academic integrity.

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!)

Here is the list of homework assignments for this course.

Here is the (outline) of solutions of homework assignments for this course.

  • Homework 1 (Due 4/4)

  • Homework 2 (Due 4/11)

  • Homework 3 (Due 4/18)

  • Homework 4 (Due 4/25)

  • Homework 5 (Due 5/2)

  • Homework 6 (Due 5/9)

  • Homework 7 (Due 5/16)

  • Homework 8 (Due 5/23) Write the solutions only to the following problems: 1(d), 1(e), 1(g), 2, 4(a), 4(c), 4(e). You still need to solve the other parts, and those parts are needed for the rest of the problems, but you do not need to write those solutions.

  • Homework 9 (Due 5/30)

  • Homework 10 (Due 6/4)