This course is a continuation of Abstract Algebra I: we will revisit structures like groups, rings, and fields as well as mappings like homomorphisms and isomorphisms.
We will also take a look at ring factorization, which will lead us to a discussion of the solutions of polynomials over abstracted structures instead of numbers sets. We will end the section on rings with a discussion of general lattices, which have both set and logical properties, and a special type of lattice known as Boolean algebra, which plays an important role in probability. We will also visit an important topic in mathematics that you have likely encountered already: vector spaces. Vector spaces are central to the study of linear algebra, but because they are extended groups, group theory and geometric methods can be used to study them.
Later in this course, we will take a look at more advanced topics and consider several useful theorems and counting methods. We will end the course by studying Galois theory—one of the most important theories in algebra, but one that requires a thorough understanding of much of the content we will study beforehand.
Upon successful completion of this course, students will be able to:
- Compute the sizes of finite groups when certain properties are known about those groups.
- Identify and manipulate solvable and nilpotent groups.
- Determine whether a polynomial ring is divisible or not and divide the polynomial (if it is divisible).
- Determine the basis of a vector space, change bases, and manipulate linear transformations.
- Define and use the Fundamental Theorem of Invertible Matrices.
- Use Galois theory to find general solutions of a polynomial over a field.
Course Requirements:
Have completed Linear Algebra, Linear Algebra II and Abstract Algebra I.
