Mathematics, Statistics & Computer Science
University of Wisconsin-Stout
Jarvis Hall Science Wing 231
Menomonie, WI 54751-0790
COURSE NUMBER/TITLE: MATH-720 Algebraic Structures
CREDITS:
3
COURSE DESCRIPTION:
Modular arithmetic, Chinese remainder theorem, linear Diophantine
equations, quadratic reciprocity, vector spaces, matrices, inner products,
quadratic forms, finite fields, field extensions and solutions to equations
over finite fields.
TEXTBOOK: TBD
COURSE OBJECTIVES:
Upon successful completion of this course the students will be able to demonstrate a mastery of the
theoretical techniques necessary to do the following:
1. Perform modular arithmetic.
2. Solve linear Diophantine equations.
3. Solve polynomial equations modulo an integer.
4. Represent linear transformations of vector spaces as matrices.
5. Perform operations on matrices including: addition/subtraction, multiplication, inversion, and
diagonalization.
6. Solve systems of linear equations using matrices.
7. Perform algebraic operations over finite fields.
8. Classify finite fields.
9. Solve systems of linear and polynomial equations over finite fields.
COURSE OUTLINE:
1. Number Theory (Objectives 1, 2, 3)
a. Modular arithmetic
b. The extended Euclidean algorithm
c. Solving linear Diophantine equations
d. The Chinese remainder theorem
e. The units modulo n
i. Fermat’s little theorem
ii. Euler’s totient function
f. Quadratic reciprocity
g. Polynomial equations modulo n
i. Quadratic forms
ii. The number of solutions to higher degree equations
2. Linear Algebra (Objectives 4, 5, 6)
a. Vector spaces
i. Formal definition
ii. Subspaces
b. Matrices
i. Morphisms of vector spaces
3.
ii. Matrix algebra, determinants, traces and inverses
iii. Systems of equations and the rank of a matrix
c. Advanced topics
i. Eigenvalues and eigenvectors
ii. Diagonalization
iii. Inner products and quadratic forms
Finite Fields (Objectives 7, 8, 9)
a. Finite fields of prime order
b. Polynomial rings over finite fields
c. Algebraic extensions of finite fields
d. Classification of finite fields
e. Select topics from linear algebra over finite fields
f. Solutions to polynomial equations over finite fields
Updated 2/2016
12/2012