Oddbean new post about login | logout @133234fb - 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?
@b4c50e1b @b4c50e1b @133234fb insane approach? really? give me a better approach to understand \( \overline{\mathbb{F}_2} \). it's a beautiful paper if you ever really want to understand Galois theory. try to prove every statement Conway made in ONAG about \( On_2 \) being the algebraic closure of \( \mathbb{F}_2 \) (or give them as exercises to your better students) Lenstra went way beyond that in the late 70ties (i think he was a visiting scholar at IHES at the time) Probably he had to hardcode things, and because of the limited computing power at the time, he could not even approach stage 47. An eternity ago, i did a post on this, slightly extending his scope (up to level 67) http://www.neverendingbooks.org/on2-extending-lenstras-list The best result I know of is by Aaron Siegel, see his book 'Combinatorial Game Theory' or my preview post on it http://www.neverendingbooks.org/aaron-siegel-on-transfinite-number-hacking He was able to push things up to level 293, but had to skip 4 levels because calculations seemed to take forever. it is quite a humbling experience that even the simplest (and in applications the most valuable) of all algebraic closures is still way beyond our reach. Later Lenstra and Bart de Smit gave another approach to the "Standard Model of Finite Fields" http://www.damtp.cam.ac.uk/user/na/FoCM/FoCM08/Talks/Lenstra.pdf