Abstract Algebra and Galois Theory

MATH 676

Overall Course Reflection

Abstract Algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. While algebra is typically thought of as solving equations, abstract algebra focuses more on the underlying structure and being able to generalize the properties. Some of the main things that we focused on were solving equations with complex numbers, set and group theory, fields and rings, and extensions into Galois groups.

What I really appreciated about this course was the connection that it had to teaching a College Algebra course for high school students. Even though many of the topics we covered go far beyond what I would touch in a College Algebra course, it is vital to understand the underlying theory behind the content. This helps me to determine the most effective ways to teach concepts and explain why certain properties and procedures work. This deeper knowledge not only strengthens my own mathematical foundation but also allows me to build stronger conceptual understanding for my students.

Course Artifacts

Solving Polynomials Presentation

This was a presentation that I created on solving polynomial equations and understanding roots. This was designed for high school students in a College Algebra class learning about polynomials. Important concepts that get covered include rational zeros theorem, roots (rational, irrational, and complex), and depressed equations.

(Creation time: 6 hours)

Proof of Solvability of Polynomials

This presentation was created to discuss an important topic from Abstract Algebra with a connection to Galois Theory. The topic that I chose to prove was why there is no formula to solve polynomials of degree $n \ge 5$ .

I begin with an overview of what solvability by radicals is, followed by theorems that are vital to use to be able to complete this proof. I then give a step-by-step overview of the proof “The Galois group $G_\mathbb{Q}(P)$ of the polynomial $P(x)=x^5-6x+3$ over $\mathbb{Q}$ is $S_5$ , which is not solvable, therefore $P(x)$ is not solvable by radicals.” Finally, I give examples of solvable and not solvable quintic polynomials.

(Creation time: 5 hours)

Exam and Homework Assignments

These are the homework assignments and exam from this course. These showcase my ability to engage with abstract algebra at a graduate level, particularly through my construction of proofs and analyzing various theorems. They demonstrate not only my understanding of abstract algebra techniques but also my ability to communicate mathematical reasoning clearly and rigorously.

(Creation time: 12-15 hours per chapter, 4 hour for exam)

Chapter 1 Homework

Chapter 2 Homework

ExamGroupTheory-Kuehn

Chapter 3 Homework

Chapter 4 Homework