Rings and Polynomials

Joseph Mellor
14 min readMay 16, 2022

Groups may allow for solvable equations, but we need more structure to even talk about polynomials. We call these structures rings.

This article is part 3 in the How to Discover Finite Fields While Bored in Class series. The image represents a finite commutative ring with five elements.

In the previous article, we established groups as the simplest algebraic structures with solvable equations and we showed that the five color group was isomorphic to the finite Abelian group of order five.

To analyze the properties of the game, we’ll need to introduce more complex algebraic structures.

Correction to the Previous Article

In the previous article, I said that we could represent all finite Abelian groups with modular arithmetic. This statement is incorrect. I’ve rewritten the article with a complete explanation, worked examples, and a proof of Lagrange’s Theorem.

Quick Note About the Series

I don’t want to weigh this article down, but I’m starting to leave series names out of the titles of articles for several reasons. See this article for more of an explanation.

Where We’re Going

The next few articles are going to be a little more abstract since I have to introduce concepts before I can talk about specific applications. Just so you don’t get lost, I’m going to tell you where we’re going. After studying the problem, I realized that we could represent each iteration with matrix multiplication. With the cutoff method, we use the matrix

where we have ones down the three central diagonals and zeros everywhere else. We call the cutoff method’s matrix a band matrix because it’s got a band down the diagonal. We also call it a tridiagonal matrix because it’s nonzero everywhere except the three central diagonals. For the wraparound method, we would use the cutoff matrix but with ones in the top right and bottom left corners set to 1.

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Joseph Mellor

BS in Physics, Math, and CS with a minor in High-Performance Computing. You can find all my articles at https://josephmellor.xyz/articles/.