Math 412 (Fall 2019)
Syllabus (the location and time listed is for section 001, but I am also teaching section 003)
Course assistant: Chris Hafke
Sections:
001: 8:30-10am in East Hall B735
002: Taught by Dr. Janet Page
003: 2:30-4pm in East Hall B743
Office hours:
Janet (in 4847 East Hall): Monday 4pm-5pm, Tuesday 1pm-2pm, Friday 1pm-2pm
Becca (in 4835 East Hall): Monday 11am-12:30am, Thursday 1:30-3:00pm
Text:
Abstract Algebra: an introduction by Thomas W. Hungerford, 3rd edition.
Note: earlier editions are OK but homework numbering and page numbers may differ. It will be your responsibility to match things up if you use an older edition.
Course description:
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. In this course we abstract the concept of the integers. What this means is we boil the integers down to selected properties, and develop algebra based on those properties. We will first consider an abstraction of the integers called rings. We will first prove the Fundamental Theorem of Algebra, then consider the basic properties of rings and ring homomorphisms. A particularly important ring will be the ring of polynomials over a field. Then we will further abstract rings to obtain another object called groups. We will study the basics of group theory including homomorphisms, symmetry groups, the symmetric group, normal subgroups, quotient groups, and group actions. As time permits, we will look at some applications of group and ring theory to RSA encryption, elliptic curves and compass/straightedge constructions.
Warning: we differ from the book by including in our definition of ring that every ring contains 1.
Course expectations:
Math 412 students are responsible for learning the material on their own through individual reading of the textbook before coming to class. You will often work together on more theoretical concepts in small groups using worksheets in class. You will be expected to work out more computational exercises on your own, which will be supplemented with some webwork when possible. You will also have a graded, written problem set due Wednesdays. Attendance is required. There will be two exams (one midterm and a final). There will be a quiz every Monday.
Invitations (blatantly copied from the Math IBL workshop 2019 organized by Drs. Nina White and Kyle Peterson):
Share responsibility for making sure all voices are heard. If you tend to have a lot to say, make sure you leave sufficient space to hear from others. If you tend to stay quiet in group discussions, challenge yourself to contribute so others can learn from you.
Understand that we are bound to make mistakes in this space, as anyone does when approaching complex tasks or learning new skills. In particular, you are invited to step outside your comfort zone!
Webwork is due every Friday at 11:59 pm.
Problem Sets
Homework 1 due September 18
Homework 2 due September 25
Homework 3 due October 2
Homework 4 due October 9
Homework 5 due October 17
Homework 6 due October 30
Homework 7 due November 6
Homework 8 due November 13
Homework 9 due November 20
Homework 10 due November 27
Homework 11 due Friday December 6
The Final will be December 17 1:30-3:30 in 1300 Chem
Review Materials:
The Midterm will be October 24 7:30-9:10pm in Angel Hall D.
Review Materials:
Mathematician presentations:
Each student is required to give a short (2-3 minute) presentation on a mathematician with whom they identify. The presentation should be an overview of the historic context of the mathematician (could be modern), the area of math and major contributions, a short biography, and the reason you chose the mathematician. Here are some places to look:
Only one student (in each section) can present on any single mathematician and you will be allowed to choose chronologically by the order in which you present. You may "claim" a mathematician a week before your presentation.
Proof writing and general writing
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 homework!
Supplement on group actions written by Karen Smith
Schedule
To read before the first class, September 4: sections 1.1 and 1.2 in the book.
The first written assignment will be an asset map and it will be due September 9. We'll talk about this the first day of class.
Wednesday, September 4: Division algorithm and the Euclidean algorithm
To (re)read before the next class: sections 1.1, 1.2, and 1.3 in the book
Monday, September 9: Fundamental Theorem of Arithmetic
Due: Asset map
To read before the next class: section 2.1 in the book
Wednesday, September 11: Congruence
Quiz 1 (see Canvas)
To read before the next class: section 2.2 and 2.3 in the book
Quiz 2 (see Canvas)
To read before the next class: section 3.1 in the book
Wednesday, September 18: Operations
Due in class: Homework 1
To read before the next class: sections 3.1 and 3.2
Monday, September 23: Rings rings rings
Quiz 3
To read before the next class: section 3.3
Wednesday, September 25: Ring homomorphisms
Due in class: Homework 2
To read before the next class: section 4.1
Monday, September 30: More rings
Quiz 4
To read before next class: finish chapter 4
Wednesday, October 2: Polynomial rings
Due in class: Homework 3
To read before the next class: section 6.3
Monday, October 7: Ideals
Quiz 5
To read before next class: section 6.2
Wednesday October 9: Quotient rings
Due in class: Homework 4
To read before the next class: reread section 6.2
Monday, October 14: Fall break! No class
Wednesday, October 16: continuing Quotient rings
Due in class: Homework 5
Monday October 21: The First Isomorphism Theorem
To read for next week: section 7.1
Wednesday October 23: Review
The Midterm will be October 24 7:30-9:10pm in Angel Hall D.
Monday, October 28: Groups
To read for next class: section 7.2
Wednesday October 30: More groups
Due in class: Homework 6
To read before next class: section 7.3
Monday November 4: Group homomorphisms
To read before next class: section 7.5
Due in class: Homework 7
Wednesday November 6: Symmetric groups
To read before next class: section 8.1
Monday November 11: Cosets
To read before next class: the supplement on group actions written by Karen Smith (we won't get to part 3 until next week)
Wednesday November 13: Groups actions
To read before next class: keep reading the supplement on group actions written by Karen Smith, make sure you take a look at part 4
Monday November 18: More group actions
To read before next class: keep reading the supplement on group actions written by Karen Smith, make sure you have read part 3
Wednesday November 20: Orbit Stabilizers
To read before next class: section 8.2
Monday November 25: Normal subgroups
To read before next class: section 8.3
Wednesday November 27: Group quotients
To read before next class: section 8.5
Monday December 2: The First Isomorphism Theorem and simple groups
To read before next class: chapter 13
Wednesday December 4: Groups and cryptography – RSA
Some elliptic curves: 1, 2 and 3
You might find this modular arithmetic calculator helpful
Monday December 9: Elliptic curves
Review for the final exam
Final exam