The class for general rth-root finding started off as a cube-root
finder before I generalised it, and in one part of the top-level
explanatory comment, I still referred to a subgroup having index 3
rather than index r.
Also, in a later paragraph, I seem to have said 'index' several times
where I meant the concept of 'rank' I defined in the previous
paragraph.
Most of them are now _mandatory_ P3 scripts, because I'm tired of
maintaining everything to be compatible with both versions.
The current exceptions are gdb.py (which has to live with whatever gdb
gives it), and kh2reg.py (which is actually designed for other people
to use, and some of them might still be stuck on P2 for the moment).
I'm going to want to use this for finding special values in elliptic
curves' ground fields.
In order to solve cubics and quartics in F_p, you have to work in
F_{p^2}, for much the same reasons that you have to be willing to use
complex numbers if you want to solve general cubics over the reals
(even if all the eventual roots turn out to be real after all). So
I've also introduced another arithmetic class to work in that kind of
field, and a shim that glues that on to the cyclic-group root finder
from the previous commit.
I'm about to want to solve quartics mod a prime, which means I'll need
to be able to take cube roots mod p as well as square roots.
This commit introduces a more general class which can take rth roots
for any prime r, and moreover, it can do it in a general cyclic group.
(You have to tell it the group's order and give it some primitives for
doing arithmetic, plus a way of iterating over the group elements that
it can use to look for a non-rth-power and roots of unity.)
That system makes it nicely easy to test, because you can give it a
cyclic group represented as the integers under _addition_, and then
you obviously know what all the right answers are. So I've also added
a unit test system checking that.
I'm about to want to expand the underlying number-theory code, so I'll
start by moving it into a file where it has room to grow without
swamping the main purpose of eccref.py.
