Tam = Pic / Sha, src = Ono. :)

I don't see the result there... that paper seems to consider the number of primitive roots within an interval of a certain type within $\FF_{p^n}$ as $n$ grows. The result there might be deeper, but it's not clear to me whether it implies the result stated by GH above. The exponential sum method is similar, as expected I guess.

Wonderful, thank you! If you know of an early reference, please let me know too. I poked around in work of Carlitz and Davenport a bit, but didn't find this statement.

With a few caveats, a simply-connected Chevalley group over a Euclidean domain or DVR is generated by its elementary root subgroups. So since the canonical projection is surjective on root subgroups (use Chinese remainder theorem), you're in good shape. I'll post this as an answer, if I have time to find the precise caveats and references.

Yes... en.wikipedia.org/wiki/Apeirogon

Thanks @nfdc23 -- I'll send Wee Teck a note to see if he knows. I don't think this is in his papers with Yu, but I'll take another look there too. And I might actually tackle p=2 sometime... so will keep Sungmun Cho in mind for the future.

I just saw the Star Wars trailer, so now I might be inclined to look for "the source" which "makes its presence felt". My previous comment was meant to give at least one source... one which I think is not considered sufficiently.

In this spirit, I'd say that carrying carries a fair amount of the complexity. For if we represent numbers in binary, and add/multiply with school algorithms, but forget about carrying, we're left in the ring ${\mathbb F}_2[X]$. And then the Riemann hypothesis and BSD (accepting finiteness of Sha) are in the bag. But sadly $1000000 + $1000000 = $0.

Thanks! I'm happy that you're enjoying the book. I created some tutorials for novice programmers to learn Python for number theory at illustratedtheoryofnumbers.com/prog.html. I hope this complements the book well and introduces basic computational number theory.