** Math 412: Abstract Algebra **

Fall 2017

Professor Karen E. Smith (Sections 1 and 2) and Professor Jack Jeffries (Section 3)

Math 412 is an introduction to abstract algebra, required for all math majors but possibly of interest also to physicists, computer scientists, and lovers of mathematics. We will begin with ring theory: our first goal is to prove the Fundamental Theorem of Algebra, about the ring you've been studying since elementary school, the integers). In the second half, we will study group theory. In addition to developing many examples, students will prove nearly all statements in this course.

**Sections:**

Section 1: 9 am MWF in B745 East Hall

Section 2: 10 am MWF in 3096 East Hall

Section 3: 9 am in Mason Hall

All sections will use the same Syllabus, do the same classwork, take the same exams, and do the same homework, regardless of instructor. Homework for all Sections is due in class on Fridays.

**Prerequisites:**
Math 217. Students are expected to know linear algebra and to have had a course in which they have been trained in rigorous proof techniques (induction, proof by contradiction, etc).

**Course Description:** This class is an introduction to the basic concepts of algebra. The topics covered are approximately Chapters 1–9 in the textbook. The class
is roughly structured as follows: we begin with a rigorous study of arithmetic of the integers (division algorithm, primes, and unique factorization, congruences, modular arithmetic) culminating with the proof Fundamental Theorem of Arithmetic. Part II is about basic properties of rings and
ring homomorphism (ideals, quotient rings, fields). Here, another important example, which shares many properties of Z, is the ring of polynomials over a field.
Finally, in part III we study the basics of group theory (groups, group homomorphisms, symmetry groups, the symmetric group,
normal subgroups, quotient groups, and group actions on sets). The parts are not evenly spaced: I is shorter than II, and II is shorter than III.

** Course Assistant: ** Yifan Wu wuyifan@umich.edu

**DAILY UPDATE:**
Section 1 and 2 Daily Update and summarizing classroom actitivies, assignments and quiz announcements.

**REQUIRED TEXT:**
Abstract Algebra: an introduction by Thomas W. Hungerford, 3rd edition (earlier editions are OK but homework numbering and page numbers may differ).

**Further readings:**
The Joy of Sets

Mathematical Hygiene

Why all rings should have a 1, an opinion piece by one of my favorite mathematicians.

**WORKSHEETS:**

Euclidean algorithm from Friday Sept 8,
Answers

Fundamental Theorem of Arithmetic from Monday Sept 11,
Answers

Congruence from Wednesday Sept 13,
Answers

Modular Arithmetic from Friday Sept 15,
Answers

Properties of Z_n from Monday Sept 18,
Answers

Two Worksheets for Wednesday Sept 20:
Finding Inverses
(
Answers )
and
Systems of Congruences
(
Answers )

Rings from Friday Sept 22,
Answers

Ring Homomorphisms from Monday Sept 25,
Answers

** Quizzes:**

Quiz 1,
Quiz 2
Quiz 3

** Problem Sets:**

Problem Set 1 due Friday Sept 15,

Problem Set 2 due Friday Sept 22,

Problem Set 3 due Friday Sept 29,

** Webwork: **
due Wednesdays

** Exams**

Midterm Wednesday evening 6:30 - 8:30 on October 18 in NA2140.

Final December 14 morning 10:30 - 12:30. The room will be announced later.

**OFFICE HOURS:**

For Karen Smith:
Mondays 11-12, Wednesdays 1-3, in my office 3074 East Hall

For Jack Jeffries: Mondays 1-3, Wednesdays 12-1, in his office 4827 East Hall

For Yifan Wu: Thursdays 5-6 in the Math Atrium.

**COURSE EXPECTATIONS:**
Math 412 students are responsible for learning the material on their own through individual reading of
the textbook ** before ** coming to class. In class, my hope was to have you working together on
more theoretical concepts in small groups using worksheets in class; however, the size of the class has grown, so we will see how this goes. You will be expected to work out more computational exercises on your own , and I will try to supplement this with some webwork if possible. You will also have a graded, written problem (think Math 217 Part B) set due Fridays. ATTENDENCE IS REQUIRED.
There will be an evening midterm (October 18: mark your calendar now) and a final (TBA).
There will be many quizzes, some on the reading.
I am available
also by appointment if you need me and can't make regular office hours.

**Grades:**
Grades will be determined as follows:

Quizzes 25%

Webwork 5%

Problem Sets (due Fridays) 15%

Midterm on October 18: 25%

Final 30%

**Testing and Disability:** If you think you need an accommodation for a disability, please let me know as soon as
possible. In particular, a Verified Individualized Services and Accommo
dations (VISA)
form must be provided to me at least
two weeks
prior to the need for a test/quiz
accommodation.
The Services for Students with Disabilities (SSD) Office
issues V
ISA forms.