MATH 3272 Abstract Algebra II
Students who take this course will require knowledge of the basic concepts of Algebra. Thus, ABSTRACT ALGEBRA I and LINEAR ALGEBRA are both listed as pre-requisites. The first part of the course continues the treatment of Groups started in ABSTRACT ALGEBRA I. Some important subgroups are defined, and the important concept of a group acting on a set is introduced. The power of group actions is demonstrated by using the technique to prove several key results about finite groups. The investigation of finite groups is concluded with the famous Sylow Theorems.
The construction of the (finite) direct product should be familiar to any mathematician, and so the course proceeds to do this. Abelian groups are discussed briefly; a statement of the Decomposition Theorem for finite groups is given. The section on Group Theory is concluded with a discussion of subgroup series – an important technique in determining the structure of a group. The Jordan-Holder Theorem is proved, and an important class of groups - the solvable groups are introduced. The course then shows focus to one of the most important examples of a Euclidean ring – the polynomial ring over a field. (Euclidean rings were introduced in ABSTRACT ALGEBRA I.) The fundamental results that transfer from Euclidean rings are restated in context, and the idea of irreducibility is introduced. The course then specialises to the rational field, and several key results concerning polynomials over the rationals are proved.
The course naturally progresses to investigate the existence of roots of polynomials over their base field. The extremely important construction of the algebraic extension containing the root of a polynomial is done in detail, with several interesting and motivating examples. The course continues to prove the existence of a splitting field, and concludes with a statement of the Fundamental Theorem of Algebra. Straightedge and compass constructions will be presented as an application if time permits.
Since cogent communication of mathematical ideas is important in the presentation of proofs, the course will emphasize clear, concise exposition. This course will therefore be useful for all students who wish to improve their skills in mathematical proof and exposition, or who intend to study more advanced topics in mathematics.