@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