# Math 542 "Modern Algebra" (Spring 2020)

Lecture Time/Place: Monday, Wednesday, Friday 12:00PM - 1:00 PM at Van Vleck B139
Lecturer: Michael Kemeny (kemeny@wisc.edu)

Course Description:
Math 542 is a second course in algebra, a central area of math. This course is necessary for those who wish to go to grad school in math.
The three core topics are 1) modules and general commutative algebra, 2) the abstract theory of matrices and 3) Galois Theory.

Text: Abstract Algebra by Dummit and Foote

Syllabus: The syllabus is available at the following link: Math542-Syllabus

Office Hours:
Monday 1:00PM - 2:00 PM in Van Vleck 319

Homework: I will assign homework from the textbook each Friday and due in class the following Friday.

Homework must be handed in at start of class. Late assignments will not be accepted.

Homework 1 (Due 02/07/20)
Our convention is that all rings are commutative with 1 (unless otherwise specified). All subrings must contain 1. Any homomorphism must send 1 to 1.
Section 7.1: 1,2,4,7,8. Section 7.3: 2,4,5,6,7,10,18
Honors: Section 7.1: 13, 14. Section 7.3 : 25, 26, 29.

Homework 2 (Due 02/14/20)
Section 10.1: 4,5,8,9. Section 10.2: 2,3,4,5,6
Honors: Section 10.1: 23. Section 10.2 : 11,14.

Homework 3 (Due 02/21/20)
Section 8.2: 3,4,5. Section 9.2: 1,2, 8, 9, 11
Honors: Section 8.2: 7. Section 9.2 : 12.

Homework 4 (Due 02/28/20)
Section 12.1: 1,2,3,5,8,11,13,14
Honors: Section 12.1: 17,18,19

Homework 5 (Due 03/6/20)
Section 8.3: 2,4,5,6,7
Honors: Section 8.3: 8, 10

Homework 6 (Due 03/13/20)
Study for Midterm!

Topics Covered:

• Modules: submodules, homomorphisms, quotients, simple modules, cyclic modules, free modules, linear transformations
• Matrices: determinants, elementary matrices, Smith normal form over a PID, characteristic roots
• Finitely generated modules over a PID
• Classification of Finite abelian groups
• Normal forms for matrices: rational canonical form, Jordan canonical form
• Field extensions, roots of polynomials