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.

Thumbs Up for Abstract Algebra! courtesy of Celia van den Bogert, Aaruran Chandrasekhar and Ana Rakovic (photo cred)**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 and videos:**
The Joy of Sets

Mathematical Hygiene

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

On the Importance of writing well, a commentary from my friend Ravi Vakil. Everything he says about the importance of writing well applies also to writing your Math 412 homeworks!

Advice on having a good life.

Public Key Cryptography
by Edward Frankel, a pretty interesting mathematician based at UC Berkeley.

What to do when the Trisector Comes
by
Underwood Dudley, who is funny in an old-timey way and happens to be the author of one of my favorite textbooks,
Elementary Number Theory.

Galois Theory
by Ian Stewart,
which is the book I would teach from if we continued with Math 413. I think you'll find it is light and easy after 412, as well as fascinating. Here, you will learn why there is not formula analogous to the quadratic formula for finding roots of polynomials of dgree at least 5. It comes down to the fact that the alternating group is a simple group (for n at least five)!

**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

More Homomorphisms from Wednesday Sept 27,
Answers

Polynomial Rings from Friday Sept 29,
Answers

More Polynomial Rings from Monday October 2,
Answers

Ideals from Wednesday October 4,
Answers

Congruence Mod I from Friday October 6,
Answers

Quotient Rings from Monday October 9,
Answers

The First Isomorphism Theorem from Wednesday October 11,
Answers

The First Isomorphism Theorem, Again from Friday October 13,
Answers

The Fundamental Theorem of Algebra from Wednesday October 18,
Answers

Groups from Friday October 20.
Answers

Groups II from Monday October 23.
Answers

LaGrange's Theorem from Wednesday October 25.
Answers

The Symmetric Group from Friday October 27.
Answers

Classifying Groups of Small Order from Monday October 30.
Answers

Homomorphisms from Wednesday Nov 1
Answers

Cosets and the Proof of Lagrange's Theorem from Friday Nov 3
Answers

Group Actions: Orbits from Monday Nov 6
Answers

Group Actions: Stabilizers from Wednesay Nov 8
Answers and Answer to Problem A
courtesy of Celia van den Bogert, Aaruran Chandrasekhar and
Ana Rakovic (photo cred)

Symmetry Groups from Friday Nov 10
Answers

Normal Subgroups from Monday Nov 13
Answers

Quotient Groups from Wednesday, November 15
Answers

Quotient Group Practice from Friday, November 17
Answers

Simple Groups and Conjugacy classes from Monday, November 20
Answers

Public Key Cryptography from Wednesday, November 22
Answers

Simple Groups from Monday November 27 and Wednesday, November 29
Answers

Classification of Finite Abelian Groups from Friday
Dec 1 Answers

Cayley's Theorem from Monday
Dec 4 Answers

Constructibility from Ancient Greece from Wednesday and Friday,
Dec 4 and 6.

The Galois Group from Monday Dec 11.

** Final Study Materials:**

Definitions Document Part 1, first half of course.

Definitions Document Part 2, second half of course, courtesy of Jin Yang.

Supplement on Group Actions, important topics not in the book. The main things to memorize here are definition of group action, orbit, stabilizer, and the orbit-stabilizer theorem. The alternative way to think of group actions in the last section is also important.

Study Questions, written by Jack Jeffries (please also review the midterm questions a bit, see below)

Some Practice T/F Questions (please also practice the midterm TF a bit, see below)

Some "Easy Proof" and Computation Practice Problems , (again, don't forget the stuff on rings from the midterm; see below)

** Midterm Study Materials:**

Definitions Document,

Study Questions,

Some Practice T/F on Chapters 4 and 5,

Some "Easy Proof" and Computation Practice Problems ,

** Quizzes:**

Quiz 1,
Quiz 2,
Quiz 3,
Quiz 4,
Quiz 5
Quiz 6
Quiz 7
Quiz 8
Quiz 9
Quiz 10
Quiz 11

** Problem Sets:**

Problem Set 1 due Friday Sept 15,

Problem Set 2 due Friday Sept 22,

Problem Set 3 due Friday Sept 29,

Problem Set 4 due Friday Oct 6,

Problem Set 5 due Friday Oct 13,

Problem Set 6 due Friday Oct 27,

Problem Set 7 due Friday Nov 3,
(
Solutions)

Problem Set 8 due Friday Nov 10,

Problem Set 9 due Friday Nov 17, (
Solutions)

Problem Set 10 due Friday Dec 1,

Problem Set 11 due Friday Dec 8,

** Webwork: **
due Wednesdays

** Exams**

Midterm Wednesday evening 6:30 - 8:30 on October 18 in NS 2140.

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.