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. JordanHolder 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 18 and 1014 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 inlibrary 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 inclass 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.
Inclass 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 makeup
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 1.11.5 of [DF], Sections 14 of [R1].
Notes.
 Aug 23

Isomorphism theorems. Section 3.3 of [DF], Sections 510 of [R1].
Notes.
 Aug 25

Free groups. Section 6.3 of [DF], Sections 1114 of [R1].
Notes.
HW 1 due.
Solutions.
 Aug 28

Group presentations; intro to group actions. Sections 6.3 and 1.7 of
[DF], Sections 1517 of [R1].
Notes.
 Aug 30

More on group actions. Section 4.1 of [DF],
Sections 1820 of [R1].
Notes.
 Sep 1

Applications of group actions, especially conjugacy classes.
Sections 4.23 of [DF], Sections 2125 of [R1]. Notes.
HW 2 due.
Solutions.
 Sep 4

Labor Day. No class.
 Sep 6

Automorphisms groups of groups; Sections 4.45 of [DF], Sections
2629 of [R1].
Notes.
 Sep 8

The Sylow Theorems; Section 4.5 of [DF], Sections 2934 of [R1].
Notes.
HW 3 due.
Solutions.
 Sep 11

Finitely generated groups and the ascending chain condition;
Section 3638 of [R1].
Notes.
 Sep 13

Torsion in abelian groups and direct products; Section 5.1 of
[DF] and Sections 3841 of [R1].
Notes.
 Sep 16

Classification of finitely generated abelian groups; Section 5.2
of [DF] and Sections 4345 of [R1].
Notes.
HW 4 due.
Solutions.
 Sep 18

Composition series and the JordanHolder theorem; Sections 5.5
and 3.4 of [DF] and Sections 4648 of [R1].
Notes.
 Sep 20

Solvable and nilpotent groups; Section 6.1 of [DF] and 4951 of [R1].
Notes.
 Sep 22

Review of rings; Sections 7.1 and 7.3 of [DF] and 16 of [R2].
Notes.
HW 5 due.
Solutions.
 Sep 25

Isomorphism theorems for rings, more examples; Sections 7.2
and 7.3 of [DF] and 710 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 1116 of [R2].
Notes.
 Oct 2

Existence of maximal ideals, rings of fractions; Sections 7.4
and 7.5 of [DF] and Sections 1721 of [R2].
Notes.
 Oct 4

Eucidean Domains amd Principal Ideal
Domains; Sections 8.18.2 of [DF] and 2226 of [R2].
Notes.
 Oct 6

Examples of nonPIDs; irreducible and prime elements; Sections
8.28.3 of [DF] and Sections 2730 of [R2].
Notes.
HW 6 due.
Solutions.
 Oct 9

PIDs are UFDs, factorization in Z[i]; Section 8.3 of [DF] and Sections 3133 of [R2].
Notes.
 Oct 11

Which polynomial rings are UFDs?; Section 9.3 of [DF] and
Sections 3437 of [R2].
Notes.
 Oct 13

Unique factorization in R[x]; irreducibility criteria; Sections
9.39.4 of [DF] and Sections 3841 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 4148 of [R2].
Notes.
 Oct 18

Module basics; Sections 10.2 and 10.3 of [DF] and Sections 4852
of [R2].
Notes.
 Oct 20

Free modules, torsion modules, Rlinear independence; Section
10.3 of [DF] and Sections 52, 5657 of [R2].
Notes.
HW 8 due.
Solutions.
 Oct 23

Classification of modules over a PID; Section 12.1 of [DF] and Sections 5861 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 13.113.2 of
[DF] and 57 of [R3].
Notes.
 Nov 6

Finite extensions; compositums of subfields. Section 13.2 of
[DF] and 69 of [R3].
Notes.
Here are some proofs of
the Fundamental Theorem of Algebra.
 Nov 8

Splitting fields and separable polynomials.
Sections 13.413.5 of [DF] and 1215 of [R3].
Notes.
 Nov 10

Finite fields and cyclotomic fields. Sections 13.513.6 of
[DF] and 23, 25, 28, 29 of [R3].
Notes.
HW 10 due.
Solutions.
 Nov 13

Introduction to Galois Theory. Section 14.114.2 of [DF].
Notes.
 Nov 15

Galois groups of splitting fields. Section 14.2 of [DF].
Notes.
 Nov 17

Foundations of Galois theory. Sections 14.23 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".