[c1ccccc1]

Why is that? My guess would be that you could adjoin an i all the time to the p^n field and get the p^2n field, as long as you had p = 4k + 3. But that's admittedly based on approximately zero thinking.

EDIT: Looking things up indicates that if n is even, there's already a square root of -1 in the field, so we can't add another. So now I believe the 1/4 of the time thing you mentioned, and can't see how that's wrong.

[dperrin]

Spitballing here, but I suspect it's a density thing. If you are considering all prime powers up to some bound N, then the density of prime powers (edit: of size p^n with n > 1) approaches 0 as N tends to infinity. So rather than things being 1/4 like our intuition says, it should unintuitively be 1/2. I haven't given this much thought, but I suspect this based on checking some examples in Sage.

[c1ccccc1]

Oh, so just a probability density thing where we sample q and check if it's p^n (retrying if not) rather than sampling p and n separately and computing q=p^n? I guess that's probably what the they were going for, yeah.

[Syzygies]

exactly