As this year is coming to a close, I would like to thank my supervisor with a gift — a story on Finite Fields (FF).

Originally this is meant to be a challenge to myself — to explain FF in my own words. I feel I should be able to explain the topic in a simple way, and so I try to put down my thoughts.

My plan was to put the thoughts on this blog, but then I encounter a problem: how much math should I assume the reader will have? Since the blog is open to the world, I don’t want to exclude anyone interested in this topic. So the assumed math level keeps going down: can’t assume abstract algebra, can’t assume set theory, can’t assume logic, etc. Very quickly I’m back to the basics: assume a novice.

Explaining a math topic to a novice is not easy, but it should be easier than explaining to a theorem-prover!

There is a bit of struggle at first, since all FF books have math so densely packed that whatever is ‘simple’ are deeply hidden. Eventually I decide to abandon the usual math book format, switching to a story-style.

This sort of suits me. When I read books on my own, there are always questions I ask (why can’t it be this or that) but no answers are forthcoming. The question remains at the back of my mind until I read another book and find the answer, but by then usually more questions have been collected without answers. I always wish there is someone nearby who will tell me, or guide me to the answer when my question is first raised.

By now I certainly have read many books on FF, so I would like to start from the beginning, ask myself questions, and try to answer them myself. By doing so, I clarify my thoughts, and gain a deeper understanding of FF. I can equip myself for my research work next year.

I am still developing the storyline. So far I think I’m on the right track.

As the story unfolds, I find that I’m going towards the basics of theorem-proving. This is quite unexpected, since only FF is in my original plan for the content. Maybe my recent reading of the HOL4 manuals is having an effect on my subconscious. Anyway, the twist to theorem-proving is quite natural in these talks of FF, which is most pleasing.

Here is the story.

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