MATHS 412: Abstract Algebra 2 (3)

Syllabus

 

1.  Prerequisite:  Maths 411 or permission of the department chairperson. 

 

2.  Course Description:  An introduction to the theory of rings, including integral domains, division rings, and fields.  Quotient fields of integral domains.  Homomorphisms, ideals, and quotient structures.  Factorization in commutative rings.  Polynomial rings and field extensions.  Aspects of Galois theory.    

 

3.  Course Objectives:  This course treats the algebraic structure known as a ring.  Students will

appreciate a unified treatment which ties together properties observed in specific examples from number systems, modular arithmetic, high school algebra, and linear algebra.  These familiar properties include the notions of factorization, root, fraction, and complex conjugation; these ideas acquire richness when studied theoretically and through new examples.  Students will observe parallels with the group theory encountered in Maths 411, such as the treatment of homomorphisms and quotient structures.  Group theory again is encountered in Galois theory, where a remarkable bijection between subgroups of a certain group and the collection of subfields of a corresponding field is explored.  Students will appreciate the usefulness of modern algebra in solving problems dating from antiquity, such as the impossibility of certain geometric constructions, and the impossibility of a general formula for roots of polynomials.

Finally, students will gain experience in writing proofs of moderate difficulty and solving problems related to the content of the course. 

 

4.  Course Rationale:  Ring theory is an important mathematical field in its own right and also a tool used in many branches of mathematics.  Consequently, Maths 412 is an essential course for students contemplating earning a graduate degree in pure mathematics, or in certain applied areas such as cryptography.  For students not expecting to study at the graduate level, Maths 411 gives exposure to a beautiful topic which provides a common structure to some mathematical objects they have encountered in high school and college.  The course primarily serves Mathematics majors, Mathematics Teaching majors, and graduate students (taught with Maths 512). 

 

5.  Course Content:  Definitions and examples of rings, including commutative and noncommutative rings, integral domains, division rings, and fields. Quotient field of an integral domain. Polynomial rings, roots, and factors. Group rings.  Homomorphisms, ideals, and quotient structures.  Prime and maximal ideals.  Principal ideal domains, unique factorization domains, and Euclidean domains.  Field extensions.  Impossibility of geometric constructions.  Finite fields.  Field automorphisms, splitting fields, Galois theory.  Some of these topics may be emphasized or de-emphasized according to the interests of the instructor and students.   

 

6.  Course Format:  Lecture/discussion.  The amount of material to be covered may not allow for a complete treatment in class of all topics listed in the Course Content, so students may need to supplement overviews in class with individual reading.      

 

7.  Methods of Evaluating Student Performance: Typical components are homework assignments, projects, and take-home exams (possibly involving group work); in-class exams; student presentations and oral exams.  The evaluation and weight of these components are at the discretion of the instructor.  Students may be evaluated on their understanding of the content material of the course, their ability to solve problems related to the course material, and on their ability to communicate mathematically to others orally and/or in writing.  

 

8.  Evaluation of the Course:  The instruction of the course is evaluated by departmental student evaluations and peer evaluation.  The course is reviewed and revised periodically by the Departmental Undergraduate Programs Committee.  

 

R. Bremigan, J. Lorch, 11/2000

V. Mascioni, 4/2004

M. Karls Spring 2006