Math 500, Abstract Algebra I
Fall 2023
Course Description
This is a graduate course in abstract algebra. The
catalog description is:
Isomorphism theorems for groups. Group actions. Composition
series. Jordan-Holder theorem. Solvable and nilpotent groups. Field
extensions. Algebraic and transcendental extensions. Algebraic
closures. Fundamental theorem of Galois theory, and
applications. Modules over commutative rings. Structure of finitely
generated modules over a principal ideal domain. Applications to
finite Abelian groups and matrix canonical forms.
with more details in the
official departmental syllabus. This corresponds roughly to
Chapters 1-8 and 10-14 of the textbook.
Prerequisites: Undergraduate linear and abstract algebra
(basics of groups, rings, fields, vectors spaces, etc.), for example as covered
in Math 416 and Math 417.
Required text: Dummit and Foote, Abstract Algebra, 3rd
Edition, 944 pages, Wiley 2003. The Grainger Engineering Library has a
copy on reserve for in-library use.
Suplemental text: Charles Rezk, Lecture Notes for Math 500,
Fall 2022: Part 1 (Groups),
Part 2 (Rings and modules),
Part 3 (Fields and Galois theory).
Course Policies
Overall grading: Your course grade will be based on
homework (40%), two in-class midterm exams (15% each), and a
comprehensive final exam (30%).
You can view all of your scores in the
online
gradebook.
Weekly homework: These will typically be due on Friday.
They are to be turned in on paper at the start of the
class period. If you are unable to attend due to illness or
quarantine, you can email me a PDF single file with a scan of your HW;
if using your phone/tablet, please use an app designed for this
purpose, such as Abobe Scan (iOS,
Android).
Late homework will not be accepted; however, your lowest two homework
grades will be dropped, so you are effectively allowed two infinitely
late assignments. Collaboration on homework is permitted, nay
encouraged! However, you must write up your solutions individually and
understand them completely.
In-class midterms: These two 50 minute exams will be held in
our usual classroom on the following Wednesdays: September 27 and
Nov 1 (weeks 6 and 11, respectively).
Final exam: The final exam will be Thursday, December 14 at
8am.
Missed exams: There will typically be no make-up
exams. Rather, in the event of a valid illness, accident, or
family crisis, you can be excused from an exam so that it does not
count toward your overall average. I reserve final judgment as to
whether an exam will be excused. All such requests should be
made in advance if possible, but in any event no more than one
week after the exam date.
Cheating: Cheating is taken very seriously as it takes
unfair advantage of the other students in the class. Penalties for
cheating on exams, in particular, are very high, typically resulting
in a 0 on the exam or an F in the class.
Disabilities: Students with disabilities who require
reasonable accommodations should see me as soon as possible. In
particular, any accommodation on exams must be requested at least a
week in advance and will require a letter from DRES.
Detailed Schedule
Includes scans of my lecture notes and the homework assignments.
Here [DF] refers to the text by Dummit and Foote, and [R1], [R2],
and [R3] refers to the three parts of Rezk's notes.
- Aug 21
-
Introduction. Sections 1.1-1.5 of [DF], Sections 1-4 of [R1].
Notes.
- Aug 23
-
Isomorphism theorems. Section 3.3 of [DF], Sections 5-10 of [R1].
Notes.
- Aug 25
-
Free groups. Section 6.3 of [DF], Sections 11-14 of [R1].
Notes.
HW 1 due.
Solutions.
- Aug 28
-
Group presentations; intro to group actions. Sections 6.3 and 1.7 of
[DF], Sections 15-17 of [R1].
Notes.
- Aug 30
-
More on group actions. Section 4.1 of [DF],
Sections 18-20 of [R1].
Notes.
- Sep 1
-
Applications of group actions, especially conjugacy classes.
Sections 4.2-3 of [DF], Sections 21-25 of [R1]. Notes.
HW 2 due.
Solutions.
- Sep 4
-
Labor Day. No class.
- Sep 6
-
Automorphisms groups of groups; Sections 4.4-5 of [DF], Sections
26-29 of [R1].
Notes.
- Sep 8
-
The Sylow Theorems; Section 4.5 of [DF], Sections 29-34 of [R1].
Notes.
HW 3 due.
Solutions.
- Sep 11
-
Finitely generated groups and the ascending chain condition;
Section 36-38 of [R1].
Notes.
- Sep 13
-
Torsion in abelian groups and direct products; Section 5.1 of
[DF] and Sections 38-41 of [R1].
Notes.
- Sep 16
-
Classification of finitely generated abelian groups; Section 5.2
of [DF] and Sections 43-45 of [R1].
Notes.
HW 4 due.
Solutions.
- Sep 18
-
Composition series and the Jordan-Holder theorem; Sections 5.5
and 3.4 of [DF] and Sections 46-48 of [R1].
Notes.
- Sep 20
-
Solvable and nilpotent groups; Section 6.1 of [DF] and 49-51 of [R1].
Notes.
- Sep 22
-
Review of rings; Sections 7.1 and 7.3 of [DF] and 1-6 of [R2].
Notes.
HW 5 due.
Solutions.
- Sep 25
-
Isomorphism theorems for rings, more examples; Sections 7.2
and 7.3 of [DF] and 7-10 of [R2].
Notes.
- Sep 27
-
Midterm the First.
Handout,
exam,
and solutions.
- Sep 29
-
Polynomial rings and kinds of ideals; Sections 7.2 and 7.4 of
[DF] and 11-16 of [R2].
Notes.
- Oct 2
-
Existence of maximal ideals, rings of fractions; Sections 7.4
and 7.5 of [DF] and Sections 17-21 of [R2].
Notes.
- Oct 4
-
Eucidean Domains amd Principal Ideal
Domains; Sections 8.1-8.2 of [DF] and 22-26 of [R2].
Notes.
- Oct 6
-
Examples of non-PIDs; irreducible and prime elements; Sections
8.2-8.3 of [DF] and Sections 27-30 of [R2].
Notes.
HW 6 due.
Solutions.
- Oct 9
-
PIDs are UFDs, factorization in Z[i]; Section 8.3 of [DF] and Sections 31-33 of [R2].
Notes.
- Oct 11
-
Which polynomial rings are UFDs?; Section 9.3 of [DF] and
Sections 34-37 of [R2].
Notes.
- Oct 13
-
Unique factorization in R[x]; irreducibility criteria; Sections
9.3-9.4 of [DF] and Sections 38-41 of [R2].
Notes.
HW 7 due.
Solutions.
- Oct 16
-
Noetherian rings and the Hilbert Basis Theorem; Modules over a ring;
Sections 9.4, 9.5, and 10.1 of [DF] and Sections 41-48 of [R2].
Notes.
- Oct 18
-
Module basics; Sections 10.2 and 10.3 of [DF] and Sections 48-52
of [R2].
Notes.
- Oct 20
-
Free modules, torsion modules, R-linear independence; Section
10.3 of [DF] and Sections 52, 56-57 of [R2].
Notes.
HW 8 due.
Solutions.
- Oct 23
-
Classification of modules over a PID; Section 12.1 of [DF] and Sections 58-61 of [R2].
Notes.
- Oct 25
-
Applications of the classification: canonical forms for
linear maps. Section 12.2 of [DF] and Section 68 of [R2].
Notes.
- Oct 27
-
More applications to linear maps. Section 12.3 of [DF] and
Section 68 of [R2].
Notes.
HW 9 due.
Solutions.
- Oct 30
-
Fields and field extensions. Section 13.1 of [DF] and 1 of
[R3].
Notes.
- Nov 1
-
Midterm the Second.
Handout.
Exam.
Solutions.
- Nov 3
-
Algebraic and transcendental elements. Sections 13.1-13.2 of
[DF] and 5-7 of [R3].
Notes.
- Nov 6
-
Finite extensions; compositums of subfields. Section 13.2 of
[DF] and 6-9 of [R3].
Notes.
Here are some proofs of
the Fundamental Theorem of Algebra.
- Nov 8
-
Splitting fields and separable polynomials.
Sections 13.4-13.5 of [DF] and 12-15 of [R3].
Notes.
- Nov 10
-
Finite fields and cyclotomic fields. Sections 13.5-13.6 of
[DF] and 23, 25, 28, 29 of [R3].
Notes.
HW 10 due.
Solutions.
- Nov 13
-
Introduction to Galois Theory. Section 14.1-14.2 of [DF].
Notes.
- Nov 15
-
Galois groups of splitting fields. Section 14.2 of [DF].
Notes.
- Nov 17
-
Foundations of Galois theory. Sections 14.2-3 of [DF].
Notes.
HW 11 due.
Solutions.
- Nov 18
- Thanksgiving Break starts.
- Nov 26
- Thanksgiving Break ends.
- Nov 27
-
Fundamental Theorem of Galois Theory I. Section 14.2 of [DF].
Notes.
- Nov 29
-
Fundamental Theorem of Galois Theory II. Section 14.2 of [DF].
Notes.
- Dec 1
-
Galois groups of polynomials. Section 14.6 of [DF].
Notes.
HW 12 due.
Solutions.
- Dec 4
-
Solving equations by radicals. Section 14.7 of [DF].
Notes.
- Dec 6
-
The end. Section 14.7 of [DF].
Notes.
- Dec 14
-
Final exam. 8am to 11am.
Handout.
Solutions to "HW13".