Oddbean

@5fb5d869 - Neat! Among other things, you're finding an element of order n in SL(2,F) where the field F is the rationals with cos(π/n) adjoined. I guess this is a subfield of the 2nth cyclotomic field: the rationals with a primitive 2nth root of unity adjoined. I'm a bit confused about why we need the factor of 2 here, but the nth cyclotomic field would only have cos(2π/n) in it.

@5fb5d869 - it does, thanks! I was overlooking the difference between SL and PSL. I'm going to call 2n "N". Now that I think about it, there's an "obvious" element of order N in SL(2,F) where F is the rationals with a primitive Nth root of unity adjoined, say ζ = exp(2πi/N) Namely, it's ζ 0 0 1/ζ People who think about rotations a lot know this is conjugate to the real matrix cos θ -sin θ sin θ cos θ where θ = 2π/N. But then the less obvious part is that's it's also conjugate to 0 1 -1 2cos θ I think there's something I still don't understand well enough, some very classical stuff about the relation between the field F and ℚ[cos θ].