I don't quite get that paper, but it mentions Conway's book On Numbers and Games, presumably for his proof that "a suitable beginnjbg segment of On_2 is an algebraic closure of the two-element subfield {0,1}".

I suppose it's not where he wrote about his notation for GF(2^n), or maybe it is. Many years ago I flipped through On Numbers and Games, but I wasn't ready then. Maybe I am now.

Anyways, Richard A. Parker seems to have put Conway's compatibility criterion into a full definition and himself named the resulting polynomials after Conway. I haven't found where Parker published about this. I'm not very good at finding publications, what do people use to find stuff? Google Scholar?

@b4c50e1b 

 Ah, there it is! Thanks for the link! I'm looking forward to catch up!

I just recently noticed that I thought for yours I knew how finite fields worked, but actually didn't.

Addition ist dead easy, there's just one field for any finite order, the primes are cyclic, prime powers come from Galois extensions, so they must be like vectors over prime fields, right? Any other way to get a field of order n is going to be isomorphic.

Well, of course not. For one, F_p^2 doesn't have twice as many elements as F_p (unless p=2). It says square, not double! The elements of F_p^n are the polynomials of degree n-1. But then that's still not all!

To write up a multiplication table we need to pick an irreducible polynomial P of degree n, so that when we multiply two elements of our field, we can use P=0 to bring down the degree of our product back to the allowed range.

And here's the kicker I totally missed for more than a decade: The labeling of F_p^n depends on our choice of P! There's even an especially nice kind of canonical polynomials for this purpose.

These are called Conway polynomials, and were invented by R. Parker. They are quite difficult to compute so current computer algebra systems use precomputed tables of them, so they can label GF(q)'s elements consistently.

@b4c50e1b 

 When I said "lesser known algebraic gadgets" I had just heard you mention rigs. You didn't go on much further that way, maybe I should have finished watching before saying silly things :D

Nice talk though. The other talk was on August 24th, and I couldn't join on Zoom at the time.

@b4c50e1b 

 I just noticed your talk appear in my rss feed and am watching last week's part one right now. I love the way you refer to lesser known algebra gadgets as motivating examples. After all these years, there still seems to be room for fresh repackagings of introductions to categories!

A bit closer to what I'm writing on right now, is there any news on your earlier talk about fields? Is the recording yet to appear, or is it just me and I missed it?

@b4c50e1b