Representations of Finite Fields

There’s little to no easy-to-use, freely available software for working with general finite fields. Let’s fix that.

Joseph Mellor
Technological Singularity

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An apple representing finite fields.
This article is part 8 in the How to Discover Finite Fields series.

Over the past few years, I’ve spent a lot of effort trying to figure out the best way to explain concepts to a wide audience, which requires that I first figure out the best way to explain it to myself. I’ve found that a few concrete examples and counterexamples that I could play with have done much more for my understanding than dozens of pages of theorems and abstractions, and so I’ve filled my articles with important examples and counterexamples.

Not every mathematician shares my view.¹ Many mathematicians don’t need examples to understand the topics, and so they write resources that don’t use examples (or only use the most trivial examples), so I have to make my own examples.²

Luckily for me, there’s a python library called sympy that has allowed me to conduct a ton of experiments for this series. It has good support for matrices, polynomials, and finite fields of prime order. It does not have support, however, for any other finite fields. Furthermore, it’s also written in python, so it can take quite a long time to do…

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Joseph Mellor
Technological Singularity

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