TODO:
[ ] copyright headers
[ ] Docstring headers for each file with description of content of file
[ ] picling elements
[ ] RR space algorithm
[ ] implement denominator in general, which is a guaranteed multiple of the denominator
only when not pid...
[ ] reduction of divisors: algorithm
[ ] polynomial factoring of any univariate poly over a non-rational function field
[ ] checking irreducibility in FunctionField_polymod constructor
[ ] pickle doctests
[ ] TestSuite(s).run() doctests
[ ] method function_field() on algebraic curves, that give back a corresponding function field object.
[ ] compute maximal separable subfield (requested by Voloch)
[ ] factoring polynomials in n variables over rational function field
[ ] x.valuation(...) [bradshaw will do this]
[ ] optimization: new implementation of the function field API, but
using multivariate poly rings in some cases; for *speed*, but with
less base rings supported...?
DONE:
[x] phi = morphism; phi(order); phi(ideal) --- won't do, since image of order or ideal often isn't one
for general morphisms
[x] a command FunctionField to make a new function field from anything.
[x] constant_base_field versus constant_field
[x] fix when asking for equation order if defining polynomial not *integral*
[x] non-monic defining polynomials
[x] do not require defining polynomial to be monic
[x] issue with *underscores* and singular:
K.<t_ba> = FunctionField(QQ)
R.<u_ba> = K[]
L.<u_ba> = K.extension(u_ba^2 - t_ba)
L.equation_order()
BOOM!
[x] isomorphisms between function fields:
K.hom([top gen, next one down, etc.])
make sure you include easy way to gens all the way down.
[x] inverses of fractional ideals
[x] factor for elements of the rational function field
[x] ideals
[x] polynomial factoring of any univariate poly over a rational function field: reduce to bivariate over constant field?
[x] conversion back and forth between a free module over base
[x] random element
[x] numerator, denominator
[x] 100% coverage
[x] matrix of element
[x] norm
[x] trace
