Abstract Algebra = Group Theory = Modern Algebra = Junior/Senior-Level Course for Mathematics Majors,
and sometimes Mathematics Education or Physics majors.
This course is one of the most difficult courses in the undergraduate curriculum, especially
the pivot towards rigorous proofs. It is often said that mathematics students
only really learn to write quality proofs from their Abstract Algebra course.
The course topics - a thorough first study of Group Theory bridging to an introductions of Rings and
Fields - are some of the most beautiful in all of mathematics.
In the table below, you should recognize the first two rows as high school algebra: solving
general linear and quadratic equations, the latter via the ubiquitous Quadratic Formula.
The third and fourth rows generalize the quadratic polynomial equation to degree 3 (cubic equation)
and degree 4 (quartic equation). These formulas, while not so easy, can be written down,
but, thankfully, high school students are not required to memorize them!
Degree
Polynomial Equation
General Solution
1
\( ax + b = 0 \)
\( x = -\frac{b}{a} \)
2
\( ax^2 + bx + c = 0 \)
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
3
\( ax^3 + bx^2 + cx + d = 0 \)
Cardano's Formula
Transform into depressed cubic:
\( t^3 + pt + q = 0 \), where:
\( t = x + \frac{b}{3a} \)
\( p = \frac{3ac - b^2}{3a^2} \),
\( q = \frac{2b^3 - 9abc + 27a^2d}{27a^3} \)
Step 1: Depress the quartic
Substitute \( x = y - \frac{b}{4a} \) to eliminate the cubic term.
This yields a depressed quartic:
\( y^4 + py^2 + qy + r = 0 \)
Step 2: Solve the resolvent cubic
Let \( z \) satisfy:
\( z^3 - \frac{p}{2}z^2 - rz + \left( \frac{pr}{2} - \frac{q^2}{8} \right) = 0 \)
Step 3: Use \( z \) to factor the quartic
Let \( \alpha = \sqrt{2z - p} \), then define:
\( y_{1,2} = \frac{-\alpha \pm \sqrt{-2z - \frac{q}{\alpha}}}{2} \)
\( y_{3,4} = \frac{\alpha \pm \sqrt{-2z + \frac{q}{\alpha}}}{2} \)
Step 4: Back-substitute
The final roots of the original equation are:
\( x = y_i - \frac{b}{4a} \), for \( i = 1,2,3,4 \)
5
\( ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0 \)
???
Much as they tried, mathematicians could not write down a general solution of the degree 5
quintic polynomial equation as a formula
involving algebraic combinations and roots (square roots, cube roots, 4th, 5th, and higher roots)
of the degree 5 polynomial coefficients.
The degree 3 and degree 4 formulas, while impressive that their discovers were able to write them down,
are not very interesting, except to know that exact solutions are possible for degree 3 and 4
polynomial equations, however complicated.
The GREAT QUESTION was:
we cannot seem to FIND a degree 5 formula - is there a formula possible and we just
have not yet been able to write the formula down? Or,
Is the something special about the degree 5 case that BLOCKS a formula from EXISTING ??
The full answer here requires TWO SEMESTERS of Abstract Algebra to tell the very amazing story
of the degree 5 formula; our DMAT 431 - Abstract Algebra course is only the first of this
two semester sequence - but we do touch upon the symmetries and the interaction of polynomial
solutions with each other under the guiding abstract formulation that is used to solve
historical mystery, and develop parallels and background to topics in relativity and string theory from
physics.
In addition to the mandatory mental strengthening challenge of learning to write (or at least get
better at) formal mathematical proofs, this course explores these introductory group theory topics
from both geometrical and computation aspects, which are rather unique to this course presentation
(and still distinct from another new and innovative approach to this subject by Dr. Matthew Macauley
in his Visual Algebra &
Visual Algebra YouTube Series, of which we are big fans!)
Using Mathematica and some interesting constructions of matrix groups, we will also make
matrix groups of fields, matrix fields, fields of matrix fields, fields of fields of
matrix fields, .... touching upon the mathematical foundations of modern cryptography.
Sample Lecture Videos for Abstract Algebra
The DMAT 431 - Computational Abstract Algebra course uses a wonderful textbook by John Scherk which bridges traditional abstract algebra to a modern
computational approach of testing and exploring ideas, concepts, theorems, and conjectures on both the hand-written, proof-based side, as well as using Mathematica to
simultaneously explore computationally. Students will learn to read an advanced math textbook, as well as drive Mathematica, via these lectures to accompany the textbook
presentation.
There are 50+ videos for the course, spanning nearly 80 hours of video content for the
course lectures
A first course in abstract algebra with emphasis on a computational approach to the subject. Topics include congruence, permutations, groups, homomorphisms, rings, fields, representations with matrices over finite fields, the Fundamental Theorem of Algebra, and solvability questions about polynomial equations, with dual emphases on traditional symbolic proofs and computational investigations.
Prerequisite
Successful completion (C- or higher) of Linear Algebra and Multivariable Calculus or equivalent, or consent of instructor.
To understand and compute congruences and modulo arithmetic
To understand and compute permutations
To understand axiomatic definition of Groups
To understand and compute examples and constructions of Groups, including representations using matrices
To understand mappings, homomorphisms, isomorphisms, and kernels
To understand and compute cyclic, symmetric, and alternating Groups
To understand and compute initial examples of Rings and Fields, both finite and infinite
To be introduced to an initial goal of group and ring theory - the insolvability of polynomial equations by roots of degree > 4 (lack of generalization of Quadratic Formula for degree 5 and higher polynomial equations)
To understand the goal of the Fundamental Theorem of Algebra
Distance Calculus students have transferred course credits to these colleges and universities:
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Distance Calculus - Student Reviews
Daniel Marasco★★★★★
Posted: Jan 13, 2020
Courses Completed: Multivariable Calculus
This course was more affordable than many, and the flexible format was terrific for me, as I am inclined to work very diligently on tasks on my own. It could be dangerous for a person who requires external discipline more, but it works well for self-starters, allowing you to prioritize when you have other pressing work. I was a full time teacher adding a math certification, and this course allowed me to master the math while working around my teaching schedule and fitting work into moments here and there when I had time. I was able to transfer the credits to Montana State University, Bozeman for my teaching internship program without a hitch. The instructors were all very helpful and patient, even when I failed to see a ridiculously simple solution on one problem after 20 emails back and forth. Overall, I was more pleased with my experience in this class than I was with any of my other 9 courses.
Transferred Credits To: Montana State University, Bozeman
Emma C.★★★★★
Posted: Jul 22, 2021
Courses Completed: Linear Algebra
This was a great course. Flexible and informative with a great professor. It's a great option if you need to fill a prerequisite fast or if you enjoy working at your own pace.
Transferred Credits To: University of Virginia
Trevor★★★★★
Posted: Jun 19, 2025
Courses Completed: Calculus I
POSITIVES:
One of the best math classes I have ever taken. The lessons made the failures of my previous professors very apparent. In a few short minutes, things that I used to struggle with just clicked. This professor is top notch and really wants you to understand how to use the material.
NEGATIVES:
The SOFTWARE is extremely frustrating. Even after taking the time to learn, there are countless glitches. You learn to work around them, and overall, the software makes the math convenient, but its failures are sorely felt throughout the course. Make sure you save often as it crashes regularly, especially with graphs.
The assignments are easy enough but some of them don't line up with the taught material. Be prepared to do some of your own independent research to get a deeper understanding of why things are the way they are.
Transferred Credits To: US Army
Ryan★★★★★
Posted: Aug 28, 2026
Courses Completed: Finite Math
I will preface this review by saying that if you are looking to take the easiest course possible, cheat through it, and learn as little as possible, this is not the course for you. Below, I will provide my uncensored opinions to this course:
First, this course uses software called LiveMath. This software can be a little bit challenging to get the hang of, but once you do get the hang of it, it's super easy. Remember that for all new things, it takes a bit to get used to it. Overall, I was highly satisfied with this course. I thought that the instructions were extremely detailed, including a video and a textbook explaining every different concept. In addition, this course offered exposure to a lot of different aspects of finite math. I think that many would agree with me on the following, that, you know, when a teacher is passionate about what they are teaching, it is so much more enjoyable, and you can definitely see that in this course. This course can be challenging at times. Maybe it'll be easy for you, maybe it'll be hard, but I think it's important to remember that the things that challenge us are, you know, the things that make us learn and the things that make us better. There's also a math chat option for if you need help or if you have any questions. Overall, I would recommend this course. I actually initially went into this course looking for an easy course to just help me get the credit requirements, but it ended up being an amazing learning experience and completely shifted my perspective on math overall. And another thing that I thought was great in this course is throughout so many different lectures, the professor would explain how each different thing can be applicable in life. And I think that, you know, that is a lot of what people think about when they are learning something. They're like, will I ever use this? Do I need to learn this? And a lot of this actually is really applicable. And I think that even if, you know, you may not be doing these math equations in your future, I think that, you know, the psychology behind these and the rules can be applied to so many different fuacets of life.
Transferred Credits To: GWU
Aiden B.★★★★★
Posted: Jul 29, 2026
Courses Completed: Calculus II, Multivariable Calculus, Differential Equations, Linear Algebra
After completing all the math offered by my highschool, I switched over to Roger Williams for Calculus II and onward. Overall, it has been a pretty great way to understand the topics online, with transferrable college credits. The exams are not terrible, and the courses cover the vast majority (if not all) of topics covered by traditional courses, with some unique ones added to. So far, I have taken four courses, with another one or two before I graduate.
The only problem with the course is the lack of pen and paper work. This shouldn't come as a crazy surprise with an online course, but there is not much emphasis on symbolic computation. Mathematica or other software does most of the heavy computation for you. The lectures usually show how to solve the problems through paper, but those skills are rarely put into practice. Literacy Sheets act as a way to do some of that (especially with the Calculus II & III course).
All in all, this isn't a huge issue. I just memorize and practice things like double integration, Laplace, ODE solving, matrix diagonalized form, row operations, etc. With a little outside work, this resolves itself. Without Roger WIlliams, I probably would not have been able to take any of these math courses in highschool.
So overall, a great choice for highschool advanced math, especially with transferrable credits.
Email: abarger@protonmail.com
Lucas L.★★★★★
Posted: Jun 25, 2026
Courses Completed: Multivariable Calculus
The professor as well as the TAs give great feedback when you need help with problems and the videos are great at explaining concepts. Return time on work is good and the work is not too much to handle.