def make_function_field(K):
r"""
Return the function field isomorphic to this field, an isomorphism,
and its inverse.
INPUT:
- ``K`` -- a field
OUTPUT: A triple `(F,\phi,\psi)`, where `F` is a rational function field,
`\phi:K\to F` is a field isomorphism and `\psi` the inverse of `\phi`.
It is assumed that `K` is either the fraction field of a polynomial ring
over a finite field `k`, or a finite simple extension of such a field.
In the first case, `F=k_1(x)` is a rational function field over a finite
field `k_1`, where `k_1` as an *absolute* finite field isomorphic to `k`.
In the second case, `F` is a finite simple extension of a rational function
field as in the first case.
.. NOTE::
this command seems to be partly superflous by now, because the residue
of a valuation is already of type "function field" whenever this makes sense.
However, even if `K` is a function field over a finite field, it is not
guaranteed that the constant base field is a 'true' finite field, and then
it is important to change that.
"""
from mclf.curves.smooth_projective_curves import make_finite_field
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.categories.function_fields import FunctionFields
from sage.rings.function_field.function_field import is_FunctionField
if is_FunctionField(K):
k = K.constant_base_field()
if k.is_finite() and hasattr(k, "base_field"):
# k seems to be finite, but not a true finite field
# we construct a true finite field k1 isomorphic to k
# and F isomorphic to K with constant base field k1
k1, phi, psi = make_finite_field(k)
if hasattr(K, "polynomial"):
# K is an extension of a rational function field
K0 = K.rational_function_field()
F0, phi0, psi0 = make_function_field(K0)
f = K.polynomial().change_ring(phi0)
F = F0.extension(f)
return F, K.hom(F.gen(), phi0), F.hom(K.gen(), psi0)
else:
F = FunctionField(k1, K.variable_name())
return F, K.hom(F.gen(), phi), F.hom(K.gen(), psi)
else:
return K, K.Hom(K).identity(), K.Hom(K).identity()
if hasattr(K, "modulus") or hasattr(K, "polynomial"):
# we hope that K is a simple finite extension of a field which is
# isomorphic to a rational function field
K_base = K.base_field()
F_base, phi_base, psi_base = make_function_field(K_base)
if hasattr(K, "modulus"):
G = K.modulus()
else:
G = K.polynomial()
R = G.parent()
R_new = PolynomialRing(F_base, R.variable_name())
G_new = R_new([phi_base(c) for c in G.list()])
assert G_new.is_irreducible(), "G must be irreducible!"
# F = F_base.extension(G_new, R.variable_name())
F = F_base.extension(G_new, 'y')
# phi0 = R.hom(F.gen(), F)
# to construct phi:K=K_0[x]/(G) --> F=F_0[y]/(G),
# we first 'map' from K to K_0[x]
phi = K.hom(R.gen(), R, check=False)
# then from K_0[x] to F_0[y]
psi = R.hom(phi_base, R_new)
# then from F_0[y] to F = F_0[y]/(G)
phi = phi.post_compose(psi.post_compose(R_new.hom(F.gen(), F)))
psi = F.hom(K.gen(), psi_base)
return F, phi, psi
else:
# we hope that K is isomorphic to a rational function field over a
# finite field
if K in FunctionFields():
# K is already a function field
k = K.constant_base_field()
k_new, phi_base, psi_base = make_finite_field(k)
F = FunctionField(k_new, K.variable_name())
phi = K.hom(F.gen(), phi_base)
psi = F.hom(K.gen(), psi_base)
return F, phi, psi
elif hasattr(K, "function_field"):
F1 = K.function_field()
phi1 = F1.coerce_map_from(K)
psi1 = F1.hom(K.gen())
F, phi2, psi2 = make_function_field(F1)
phi = phi1.post_compose(phi2)
psi = psi2.post_compose(psi1)
return F, phi, psi
else:
raise NotImplementedError
