### Basic information

Welcome to Math 412!

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.

Warning: we differ from the book by including in our definition of ring that every ring contains 1.

Sections 1 and 2 are taught by Dr. Eloísa Grifo

Review of proof techniques

Department of Mathematics | University of Michigan | East Hall | 530 Church Street | Ann Arbor, MI 48109

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.

Warning: we differ from the book by including in our definition of ring that every ring contains 1.

**Time and place:**East Hall B735 (basement), Tuesday and Thursday 11:30am–1pm**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).**Required text:***Abstract Algebra: an introduction*by Thomas W. Hungerford, 3rd edition (earlier editions are OK but homework numbering and page numbers may differ).**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, you will work together on more theoretical concepts in small groups using worksheets; this is an essential part of the course and your grade. The course is run "IBL" style, similar to Math 217, so be prepared every time! You will be expected to work out more computational exercises on your own, which will be tested by weekly webwork. You will also have a Quiz every Tuesday, a graded, written problem (think Math 217 Part B) set due Thursdays, and Webwork due every Friday. Attndance is required.**Office hours:**In Jack's office, EH 4827, Tuesday, Wednesday, Thursday 1–2pm (subject to change). For the first week, Wednesday, Thursday, Friday 1–2pm.**Sections:**All sections will use the same Syllabus, do the same classwork, have the same webwork, take the same exams, and do the same homework, regardless of instructor. You are welcome to attend either instructor's office hours.Sections 1 and 2 are taught by Dr. Eloísa Grifo

**Webwork:**Webwork is due every Friday at 11:59 pm.**Further readings and videos:**On the Importance of writing well, a commentary from Ravi Vakil. Everything he says about the importance of writing well applies also to writing your Math 412 homeworks!### Worksheets and announcements

- Thursday, January 10: Division algorithm Solutions and the Euclidean algorithm Solutions

To (re)read before the next class: sections 1.1, 1.2, and 1.3 in the book - Tuesday, January 15: Fundamental Theorem of Arithmetic Solutions

To read before the next class: section 2.1 in the book - Thursday, January 17: Congruence Solutions

Quiz #1 solutions

To read before the next class: section 2.2, 2.3 in the book - Tuesday, January 22: Arithmetic in Zn
Solutions

Quiz #2 solutions

Midterm availability

To read before the next class: section 3.1 in the book - Thursday, January 24:
Operations
Solutions

Homework #1 Solutions

To read before the next class: 3.1 (again), and start 3.2 - Tuesday, January 29: Ring basics
Solutions

Quiz #3 Solutions

To read before next class 3.2 and 3.3 - Thursday, January 31: Snow day!
- Tuesday, February 5: Ring Homomorphisms Solutions

Quiz #4 solutions

Homework #2 Solutions

To read for next class: 4.1 - Thursday, February 7: More Rings solutions

Homework #3 solutions

To read for next class: finish chapter 4 - Tuesday, February 12: Polynomial Rings solutions

Quiz #5 solutions

To read for next class: 6.1 (yes, 6) - Thursday, February 14: Ideals solutions

Homework #4 solutions

To read for next class: 6.2 - Tuesday, February 19: Quotient rings solutions

Quiz #6 solutions

Final exam availablility - Thursday, February 21: Quotient rings solutions

Homework #5 solutions

- Tuesday, February 26: Exam 1, 6 pm, CHEM 1400

Some common mistakes

some review questions

Review questions from class solutions

- Thursday, February 28: First Isomorphism Theorem solutions

Over the break: Read section 7.1, HW #6 (short) will be collected the Thursday the week after break - Tuesday, March 12: Groups solutions

Read section 7.2 for Thursday - Thursday, March 14: Groups 2 solutions

Quiz #7 solutions

Homework #6 solutions - Tuesday, March 19: Group homomorphisms solutions

Quiz #8 solutions

Finish chapter 7 for next class - Thursday, March 21: Symmetric Groups solutions

Homework #7 solutions

Read Section 8.1 for next class - Tuesday, March 26: Cosets solutions

Quiz #9 solutions

Read the supplement on Group Actions. Focus on sections 1, 2, and 4. - Thursday, March 28: Group Actions solutions

Homework #8 solutions

- Tuesday April 2: Orbit Stabilizer Theorem
solutions

Quiz #10 solutions

Read Section 8.2 for Thursday - Thursday, April 4: Normal Subgroups
solutions

Homework #9 solutions

Read Section 8.3 for Tuesday - Tuesday, April 9: Quotient Groups
solutions

Quiz #11 solutions

Read 8.4 and 8.5 for next class - Thursday, April 11: First Isomorphism Theorem and Simple Groups
solutions

Homework #10 solutions

Read Chapter 13 for next class

- Tuesday, April 16: RSA
solutions and/or Elliptic Curves
solutions

Quiz #12 solutions

Read Chapter 15 for next class

Reminder: Fill out course evaluations

- Thursday, April 18: Constructibility
solutions

Compass and straightedge construction game

Homework #11 solutions

- Tuesday, April 23: T/F Review solutions
- Final Exam: April 25, 4–6pm, EH 1360

Final exam solutions

some review questions (updated 4/17 11am)

**Alternatives to Math 412**:

**Math 312**also satisfies the algebra requirement for the math major. This course covers much of the same material but demands a bit less in terms of what you are expected to be able to prove. It might be a better option for you if you do not like proofs, struggle with or have not had a good introduction to proofs like Math 217. Math 312 will cover some proof techniques that we will assume in Math 412.

**Math 217**If you haven't had this, take it! Math 217 is not just "matrix algebra"— it is more theoretical. It will teach you how to "do proofs" for future math classes. It is a great class, with applications all over science, engineering, and math, and the perfect prereq for Math 412.

**Math 490**This is a different topic (topology) but also moves at a similar pace and is taught in a similar way. If you are looking for an upper level math elective and don't need an "algebra" course, this is another option.

**Math 493**satisfies the algebra requirement, and is also an introduction to Abstract Algebra but it assumes students have had a much deeper introduction to abstract mathematics, such as Math 295

Review of proof techniques