Math 417 - Fall 2023

Intro to Abstract Algebra

Dan Berwick-Evans (danbe at illinois)

Office hours:

Wednesday 1:30-2:20pm in 365 Altgeld
Friday 11:00-11:50am in 365 Altgeld
Other times by appointment.

Lectures:
Section C13 10:00AM - 10:50AM MWF 149 Henry Administration Bldg

Detailed schedule, lecture notes, and assignments.

Course info

Overview: Algebra is the branch of mathematics which studies equations, their solutions, and ways to manipulate them (the word algebra comes from Arabic and means "reunion of broken parts"). The abstraction and distillation of ideas over time has led us to the definition of a group, the basic object of study in modern abstract algebra. Other central objects which we will study are rings and fields.

This is a first course in abstract algebra. Topics covered include:

Prerequisites: Either MATH 416 or one of ASRM 406, MATH 415 together with one of MATH 347, MATH 348, CS 374; or consent of instructor.

Text: Algebra: Abstract and Concrete (edition 2.6) by Goodman. This book is available free of charge here. I will be using Prof. Eugene Lerman's lecture notes, which roughly follow this text. (See the detailed schedule for how the lecture notes and textbook line up.)

Canvas site: https://canvas.illinois.edu. You can check your grades or find solutions to homework and exams here.

Course Policies

Grades: Your course grade is determined by the following:

Homework: Mathematics (and problem solving in general) is a collaborative discipline. You are strongly encouraged to work together on homework assignments. However, you must write-up the solution on your own and in your own words. Anything else is plagiarism and will be treated as such.

Homework is due before class, typically on Friday. Please submit your homework via Canvas; this should be a single pdf file and you should use an app designed for this purpose (e.g., Adobe Scan). Late homework is not accepted for any reason. However to compensate for this, your two lowest homework grades will be dropped.

Exams: There will be two in-class midterm exams and a final exam.

Exam 1: September 20.
Exam 2: November 1.
Final Exam: 8:00am-11:00am., Thursday Dec. 14.

Academic Integrity: Cheating and other forms of academic dishonesty are taken very seriously. Any violation of the Illinois Academic Integrity Policy will result in a significant penalty.

Accommodations: To obtain disability-related accommodations, students should contact both me and the Disability Resources and Educational Services (DRES) as soon as possible.

Detailed Schedule, Lecture Notes, and Assignments.

Check back regularly; lecture notes, assignments, and other files will be added throughout the semester. The schedule of topics may change as the semester progresses.

Week 1: Suggested reading: 1.1-1.5, Review Appendix A, Appendix B. (The lectures do not exactly follow the book, so the suggested reading may not always exactly align.)
August 21
Definition of a group, examples, the group of rigid motions. Lecture notes.
August 23
Symmetries of a square, permutations. Lecture notes.
August 25
Permutations, decomposing permutations into disjoint cycles. Lecture notes.
Week 2: Suggested reading: 1.5-1.6
August 28
Existence and uniqueness of the decomposition of permutations into disjoint cycles, divisibility of integers. Lecture notes.
August 30
Division algorithm, existence of gcd's. Lecture notes.
September 1
Computing gcd's, factorization into primes. Lecture notes.
Due: HW 1.
Week 3: Suggested reading: 1.6-1.7
September 4
Labor Day (no class).
September 6
Uniqueness of factorization into primes, integers mod n, equivalence relations and classes, modular arithmetic. Lecture notes.
September 8
Applications of modular arithmetic, zero divisors and units in Z_n. Lecture notes.
Due: HW 2.
Week 4: Suggested reading: 1.8, 2.1-2.2
September 11
Polynomials, division "algorithm" for polynomials. Lecture notes.
September 13
Division "algorithm" for polynomials, roots of polynomials, back to groups. Lecture notes.
September 15
Homomorphisms, subgroups, cyclic (sub)groups. Lecture notes.
Due: HW 3.
Week 5: Suggested reading: 2.2, 2.4
September 18
Review for the first midterm.
Info and review.   Solutions to review problems.
September 20
Exam 1.
September 22
Subgroups of Z, kernel, kernel measures injectivity, kernel is a normal subgroup. Lecture notes.
Week 6: Suggested reading: 2.3, 2.5-2.7
September 25
Left and right cosets. Lagrange's theorem. Lecture notes.
September 27
Quotient groups. Lecture notes.
September 29
Review of complex numbers, the dihedral group in terms of complex numbers. Lecture notes.
Due: HW 4.
Week 7: Suggested reading: 2.7, 5.1-5.2
October 2
Homomorphism theorem and some applications. Lecture notes.
October 4
Euler's theorem, Chinese remainder theorem, group actions. Lecture notes.
October 6
Another view of group actions, orbits, orbits form a partition. Lecture notes.
Due: HW 5.
Week 8: Suggested reading: 5.3-5.4
October 9
The permutation representation of the symmetric group, sign homomorphism. Lecture notes.
October 11
Stabilizers, Gx ~ G/G_x, conjugation and conjugacy classes, conjugacy classes of r-cycles in S_n. Lecture notes.
October 13
The class equation, classification of groups of order p^2 (p prime). Lecture notes.
Due: HW 6.
Week 9: Suggested reading: 3.1-3.2, 3.6
October 16
Cauchy's theorem. Lecture notes.
October 18
Semi-direct products. Lecture notes.
October 20
Sylow theorems. Lecture notes.
Due: HW 7.
Week 10: Suggested reading: 6.1-6.2
October 23
Definitions of rings, subrings, unity, units, fields, ring homomorphisms. Lecture notes.
October 25
Ideals in a field, homomorphisms out of a field, image of a homomorphism is a subring,"substitution principle," polynomials are not functions. Lecture notes.
October 27
Principal ideals, intersection of ideals, ideals generated by a set, sum of ideals. Lecture notes.
Due: HW 8.
Week 11: Suggested reading: 6.3
October 30
Review for the second midterm. Notes.
November 1
Exam 2.
Info and review.  
November 3
Products of ideals, sums of rings, quotient rings. Lecture notes.
Week 12: Suggested reading: 6.3
November 6
Homomorphism theorem for rings. Lecture notes.
November 8
Images and preimages of ideals, maximal ideals, M maximal in R iff R/M is a field. Lecture notes.
November 10
M maximal in R iff R/M is a field; integral domains, finite integral domains are fields; prime ideals, P in R prime iff R/P is an integral domain. Lecture notes.
Due: HW 9.
Week 13: Suggested reading: 6.4-6.5
November 13
Irreducibles in integral domains, PIDs, in a PID irreducible = prime and the corresponding ideal is maximal. Lecture notes.
November 15
A field with 4 elements, an integral domain where irreducibles are not primes. Lecture notes.
November 17
"review" of vector spaces over a field. For a field F and a polynomial f in F[x] of degree n>0 the quotient ring F[x]/(f) is a vector space over F of dimension n. Lecture notes.
Due: HW 10.
Fall Break!
Week 14: Suggested reading: 6.5-6.6
November 27
Characteristic of a ring, the number of elements in a finite fields is a power of a prime, subfields and extension of fields, field extensions add roots of polynomials. Lecture notes.
November 29
UFD, ascending chains of ideals in a PID, PIDs are UFDs. Lecture notes.
December 1
Field extensions, algebraic elements. Lecture notes.
Week 15: Suggested reading: 7.1-7.3
December 4
Field extensions and ruler and compass constructions. Lecture notes.
Due: HW 11.
December 6
Review for the final.
December 8
Reading day.
December 14
Final exam, 8-11am Info and review.