Ah, there it is! Thanks for the link! I'm looking forward to catch up!
I just recently noticed that I thought for years I knew how finite fields worked, but actually didn't.
Addition ist dead easy, there's just one field for any finite order, for primes they are cyclic, prime powers come from Galois extensions, so they must be like... errr... vectors over prime fields, right? After all, any other way to get a field of order n is going to be isomorphic.
Well, of course not. For one, 𝐹ₚ² doesn't have twice as many elements as 𝐹ₚ (unless p=2). It says square, not double! The elements of 𝐹ₚⁿ 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 𝐹ₚⁿ 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.
