@RefurioAnachro - I know Conway invented a decent system for naming elements of 𝐹ₚⁿ when p = 2, but now I can't find any references to this! I gues Parker generalized it.
I found this insane approach, though:
https://www.math.leidenuniv.nl/~hwl/PUBLICATIONS/1977e/art.pdf
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?
@johncarlosbaez