Upper Division Mathematics: Computational Abstract Algebra

Course Overview

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: \( t = x + \frac{b}{3a} \) \( p = \frac{3ac - b^2}{3a^2} \), \( q = \frac{2b^3 - 9abc + 27a^2d}{27a^3} \) Then: \( x = \sqrt[3]{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} - \frac{b}{3a} \)
4	\( ax^4 + bx^3 + cx^2 + dx + e = 0 \)	Ferrari's Method (Full Steps) Given: \( ax^4 + bx^3 + cx^2 + dx + e = 0 \) Step 1: Depress the quartic Substitute \( x = y - \frac{b}{4a} \) to eliminate the cubic term. This yields a depressed quartic: \( y^4 + py^2 + qy + r = 0 \) where: \( p = \frac{8ac - 3b^2}{8a^2} \) \( q = \frac{b^3 - 4abc + 8a^2d}{8a^3} \) \( r = \frac{-3b^4 + 256a^3e - 64a^2bd + 16ab^2c - 16a^2c^2}{256a^4} \) 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

Video: Introduction to Abstract Algebra

Video: Permutation Groups

Video: Linear Groups

Video: Groups

Video: Subgroups

Video: Group Actions

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Roger Williams University Course Catalog Listing: DMAT 431 - Computational Abstract Algebra

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DMAT 431 - Computational Abstract Algebra - Learning Outcomes

• 1. To understand and compute congruences and modulo arithmetic
• 2. To understand and compute permutations
• 3. To understand axiomatic definition of Groups
• 4. To understand and compute examples and constructions of Groups, including representations using matrices
• 5. To understand mappings, homomorphisms, isomorphisms, and kernels
• 6. To understand and compute cyclic, symmetric, and alternating Groups
• 7. To understand and compute initial examples of Rings and Fields, both finite and infinite
• 8. 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)
• 9. To understand the goal of the Fundamental Theorem of Algebra

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DMAT 431 - Computational Abstract Algebra - Syllabus of Topics

1.	Getting Started
	1.1	Email and Chat
	1.2	Learning About the Course
	1.3	Required Hardware
	1.4	Software Fundamentals: Mathematica

2.	Initial Questions About Polynomial Equations
	2.1	Quadratic, Cubic, Quartic Formulas
	2.2	Historical Lack of Quintic Formula
	2.3	Closer Look at Polynomial Factorizations
	2.4	The Question of Coefficients
	2.5	The Fundamental Theorem of Algebra

3.	Congruences
	3.1	Basic Properties
	3.2	Divisibility
	3.3	Modulo Arithmetic
	3.4	Solving Congruences
	3.5	The Role of Prime Numbers

4.	Permutations
	4.1	Mappings
	4.2	Cycles
	4.3	Signs of Permutations

5.	Permutation and Linear Groups
	5.1	Definition
	5.2	Cyclic Groups
	5.3	Generators

6.	General Groups
	6.1	Axiomatic Definition
	6.2	Properties
	6.3	Homomorphisms

7.	Subgroups
	7.1	Orthogonal Groups
	7.2	Subgroups and Generators
	7.3	Kernel and Image of Homomorphisms

8.	Groups of Symmetries
	8.1	Symmetries of Regular Polygons
	8.2	Symmetries of Platonic Solids
	8.3	Symmetries of Equations

9.	Group Actions
	9.1	Orbits and Stabilizers
	9.2	Fractional Linear Transformations
	9.3	Kernels and Cayley's Theorem

10.	Counting Formulas
	10.1	The Class Equations
	10.2	Burnside's Counting Lemma

11.	Simple, Sylow, and Abelian Groups
	11.1	Finite Fields and Matrix Representations 
	11.2	Group Examples

12.	Brief Introduction to Rings and Fields
	12.1	Definitions and Examples
	12.2	Questions about Polynomial Factorization
	12.3	Polynomial Equations Revisited 
	12.4	Fundamental Theorem of Algebra

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