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hom_finite_field.pyx
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hom_finite_field.pyx
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"""
Finite field morphisms
This file provides several classes implementing:
- embeddings between finite fields
- Frobenius isomorphism on finite fields
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
Construction of an embedding::
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K)); f
Ring morphism:
From: Finite Field in t of size 3^7
To: Finite Field in T of size 3^21
Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
sage: f(t)
T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
The map `f` has a method ``section`` which returns a partially defined
map which is the inverse of `f` on the image of `f`::
sage: g = f.section(); g
Section of Ring morphism:
From: Finite Field in t of size 3^7
To: Finite Field in T of size 3^21
Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
sage: g(f(t^3+t^2+1))
t^3 + t^2 + 1
sage: g(T)
Traceback (most recent call last):
...
ValueError: T is not in the image of Ring morphism:
From: Finite Field in t of size 3^7
To: Finite Field in T of size 3^21
Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
There is no embedding of `GF(5^6)` into `GF(5^11)`::
sage: k.<t> = GF(5^6)
sage: K.<T> = GF(5^11)
sage: FiniteFieldHomomorphism_generic(Hom(k, K))
Traceback (most recent call last):
...
ValueError: No embedding of Finite Field in t of size 5^6 into Finite Field in T of size 5^11
Construction of Frobenius endomorphisms::
sage: k.<t> = GF(7^14)
sage: Frob = k.frobenius_endomorphism(); Frob
Frobenius endomorphism t |--> t^7 on Finite Field in t of size 7^14
sage: Frob(t)
t^7
Some basic arithmetics is supported::
sage: Frob^2
Frobenius endomorphism t |--> t^(7^2) on Finite Field in t of size 7^14
sage: f = k.frobenius_endomorphism(7); f
Frobenius endomorphism t |--> t^(7^7) on Finite Field in t of size 7^14
sage: f*Frob
Frobenius endomorphism t |--> t^(7^8) on Finite Field in t of size 7^14
sage: Frob.order()
14
sage: f.order()
2
Note that simplifications are made automatically::
sage: Frob^16
Frobenius endomorphism t |--> t^(7^2) on Finite Field in t of size 7^14
sage: Frob^28
Identity endomorphism of Finite Field in t of size 7^14
And that comparisons work::
sage: Frob == Frob^15
True
sage: Frob^14 == Hom(k, k).identity()
True
AUTHOR:
- Xavier Caruso (2012-06-29)
"""
#############################################################################
# Copyright (C) 2012 Xavier Caruso <xavier.caruso@normalesup.org>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# http://www.gnu.org/licenses/
#****************************************************************************
from sage.rings.integer cimport Integer
from sage.categories.homset import Hom
from sage.structure.element cimport Element
from sage.rings.finite_rings.finite_field_base import is_FiniteField
from sage.rings.morphism cimport RingHomomorphism, RingHomomorphism_im_gens, FrobeniusEndomorphism_generic
from sage.rings.finite_rings.finite_field_constructor import FiniteField
from sage.categories.map cimport Section
from sage.categories.morphism cimport Morphism
from sage.misc.cachefunc import cached_method
cdef class SectionFiniteFieldHomomorphism_generic(Section):
"""
A class implementing sections of embeddings between finite fields.
"""
cpdef Element _call_(self, x): # Not optimized
"""
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: g = f.section()
sage: g(f(t^3+t^2+1))
t^3 + t^2 + 1
sage: g(T)
Traceback (most recent call last):
...
ValueError: T is not in the image of Ring morphism:
From: Finite Field in t of size 3^7
To: Finite Field in T of size 3^21
Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
"""
for root, _ in x.minimal_polynomial().roots(ring=self.codomain()):
if self._inverse(root) == x:
return root
raise ValueError("%s is not in the image of %s" % (x, self._inverse))
def _repr_(self):
"""
Return a string representation of this section.
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: g = f.section()
sage: g._repr_()
'Section of Ring morphism:\n From: Finite Field in t of size 3^7\n To: Finite Field in T of size 3^21\n Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T'
"""
return "Section of %s" % self._inverse
def _latex_(self):
r"""
Return a latex representation of this section.
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: g = f.section()
sage: g._latex_()
'\\verb"Section of "\\Bold{F}_{3^{7}} \\hookrightarrow \\Bold{F}_{3^{21}}'
"""
return '\\verb"Section of "' + self._inverse._latex_()
cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
"""
A class implementing embeddings between finite fields.
"""
def __init__(self, parent, im_gens=None, base_map=None, check=True, section_class=None):
"""
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K)); f
Ring morphism:
From: Finite Field in t of size 3^7
To: Finite Field in T of size 3^21
Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
sage: k.<t> = GF(3^6)
sage: K.<t> = GF(3^9)
sage: FiniteFieldHomomorphism_generic(Hom(k, K))
Traceback (most recent call last):
...
ValueError: No embedding of Finite Field in t of size 3^6 into Finite Field in t of size 3^9
sage: FiniteFieldHomomorphism_generic(Hom(ZZ, QQ))
Traceback (most recent call last):
...
TypeError: The domain is not a finite field
sage: R.<x> = k[]
sage: FiniteFieldHomomorphism_generic(Hom(k, R))
Traceback (most recent call last):
...
TypeError: The codomain is not a finite field
"""
domain = parent.domain()
codomain = parent.codomain()
if not is_FiniteField(domain):
raise TypeError("The domain is not a finite field")
if not is_FiniteField(codomain):
raise TypeError("The codomain is not a finite field")
if domain.characteristic() != codomain.characteristic() or codomain.degree() % domain.degree() != 0:
raise ValueError("No embedding of %s into %s" % (domain, codomain))
if im_gens is None:
im_gens = domain.modulus().any_root(codomain)
check=False
RingHomomorphism_im_gens.__init__(self, parent, im_gens, base_map=base_map, check=check)
if section_class is None:
self._section_class = SectionFiniteFieldHomomorphism_generic
else:
self._section_class = section_class
def __copy__(self):
"""
Return a copy of this map.
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: g = copy(f)
sage: g.section()(g(t)) == f.section()(f(t))
True
::
sage: F = GF(2)
sage: E = GF(4)
sage: phi = E.coerce_map_from(F); phi
Ring morphism:
From: Finite Field of size 2
To: Finite Field in z2 of size 2^2
Defn: 1 |--> 1
sage: phi.section()
Section of Ring morphism:
From: Finite Field of size 2
To: Finite Field in z2 of size 2^2
Defn: 1 |--> 1
"""
cdef FiniteFieldHomomorphism_generic out = super(FiniteFieldHomomorphism_generic, self).__copy__()
out._section_class = self._section_class
return out
def _latex_(self):
r"""
Return a latex representation of this embedding.
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: f._latex_()
'\\Bold{F}_{3^{7}} \\hookrightarrow \\Bold{F}_{3^{21}}'
"""
return self.domain()._latex_() + " \\hookrightarrow " + self.codomain()._latex_()
cpdef Element _call_(self, x):
"""
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^3)
sage: K.<T> = GF(3^9)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: f(t)
2*T^6 + 2*T^4 + T^2 + T
sage: a = k.random_element()
sage: b = k.random_element()
sage: f(a+b) == f(a) + f(b)
True
sage: f(a*b) == f(a) * f(b)
True
"""
f = x.polynomial()
bm = self.base_map()
if bm is not None:
f = f.map_coefficients(bm)
return f(self.im_gens()[0])
def is_injective(self):
"""
Return True since a embedding between finite fields is
always injective.
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^3)
sage: K.<T> = GF(3^9)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: f.is_injective()
True
"""
return True
def is_surjective(self):
"""
Return true if this embedding is surjective (and hence an
isomorphism.
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^3)
sage: K.<T> = GF(3^9)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: f.is_surjective()
False
sage: g = FiniteFieldHomomorphism_generic(Hom(k, k))
sage: g.is_surjective()
True
"""
return self.domain().cardinality() == self.codomain().cardinality()
@cached_method
def section(self):
"""
Return the ``inverse`` of this embedding.
It is a partially defined map whose domain is the codomain
of the embedding, but which is only defined on the image of
the embedding.
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: g = f.section(); g
Section of Ring morphism:
From: Finite Field in t of size 3^7
To: Finite Field in T of size 3^21
Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
sage: g(f(t^3+t^2+1))
t^3 + t^2 + 1
sage: g(T)
Traceback (most recent call last):
...
ValueError: T is not in the image of Ring morphism:
From: Finite Field in t of size 3^7
To: Finite Field in T of size 3^21
Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
"""
if self.base_map() is not None:
raise NotImplementedError
return self._section_class(self)
def inverse_image(self, b):
"""
Return the unique ``a`` such that ``self(a) = b`` if one such exists.
This method is simply a shorthand for calling the map returned by
``self.section()`` on ``b``.
EXAMPLES::
sage: k.<t> = GF(3^7)
sage: K.<T>, f = k.extension(3, map=True)
sage: b = f(t^2); b
2*T^20 + 2*T^19 + T^18 + T^15 + 2*T^14 + 2*T^13 + 2*T^12 + T^8 + 2*T^6 + T^5 + 2*T^4 + T^3 + 2*T^2 + T
sage: f.inverse_image(b)
t^2
sage: f.inverse_image(T)
Traceback (most recent call last):
...
ValueError: T is not in the image of Ring morphism:
From: Finite Field in t of size 3^7
To: Finite Field in T of size 3^21
Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
"""
return self.section()(b)
def __hash__(self):
return Morphism.__hash__(self)
cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
"""
A class implementing Frobenius endomorphisms on finite fields.
TESTS::
sage: k.<a> = GF(7^11)
sage: Frob = k.frobenius_endomorphism(5)
sage: TestSuite(Frob).run()
"""
def __init__(self, domain, n=1):
"""
INPUT:
- ``domain`` -- a finite field
- ``n`` -- an integer (default: 1)
.. NOTE::
`n` may be negative.
OUTPUT:
The `n`-th power of the absolute (arithmetic) Frobenius
endomorphism on ``domain``
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FrobeniusEndomorphism_finite_field
sage: k.<t> = GF(5^3)
sage: FrobeniusEndomorphism_finite_field(k)
Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^3
sage: FrobeniusEndomorphism_finite_field(k, 2)
Frobenius endomorphism t |--> t^(5^2) on Finite Field in t of size 5^3
sage: FrobeniusEndomorphism_finite_field(k, t)
Traceback (most recent call last):
...
TypeError: n (=t) is not an integer
sage: FrobeniusEndomorphism_finite_field(k['x'])
Traceback (most recent call last):
...
TypeError: The domain must be a finite field
"""
if not is_FiniteField(domain):
raise TypeError("The domain must be a finite field")
try:
n = Integer(n)
except TypeError:
raise TypeError("n (=%s) is not an integer" % n)
self._degree = domain.degree()
self._power = n % self._degree
self._degree_fixed = domain.degree().gcd(self._power)
self._order = self._degree / self._degree_fixed
self._q = domain.characteristic() ** self._power
RingHomomorphism.__init__(self, Hom(domain, domain))
def _repr_(self):
"""
Return a string representation of this endomorphism.
EXAMPLES::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism(); Frob
Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^3
sage: Frob._repr_()
'Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^3'
"""
name = self.domain().variable_name()
if self._power == 0:
s = "Identity endomorphism of"
elif self._power == 1:
s = "Frobenius endomorphism %s |--> %s^%s on" % (name, name, self.domain().characteristic())
else:
s = "Frobenius endomorphism %s |--> %s^(%s^%s) on" % (name, name, self.domain().characteristic(), self._power)
s += " %s" % self.domain()
return s
def _repr_short(self):
"""
Return a short string representation of this endomorphism.
EXAMPLES::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism(); Frob
Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^3
sage: Frob._repr_short()
't |--> t^5'
"""
name = self.domain().variable_name()
if self._power == 0:
s = "Identity"
elif self._power == 1:
s = "%s |--> %s^%s" % (name, name, self.domain().characteristic())
else:
s = "%s |--> %s^(%s^%s)" % (name, name, self.domain().characteristic(), self._power)
return s
def _latex_(self):
r"""
Return a latex representation of this endomorphism.
EXAMPLES::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: Frob._latex_()
't \\mapsto t^{5}'
"""
try:
name = self.domain().latex_variable_names()[0]
except IndexError:
name = "x"
if self._power == 0:
s = '\\verb"id"'
elif self._power == 1:
s = "%s \\mapsto %s^{%s}" % (name, name, self.domain().characteristic())
else:
s = "%s \\mapsto %s^{%s^{%s}}" % (name, name, self.domain().characteristic(), self._power)
return s
cpdef Element _call_(self, x):
"""
TESTS::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: Frob(t)
2*t^2 + 4*t + 4
sage: Frob(t) == t^5
True
"""
return x ** self._q
def order(self):
"""
Return the order of this endomorphism.
EXAMPLES::
sage: k.<t> = GF(5^12)
sage: Frob = k.frobenius_endomorphism()
sage: Frob.order()
12
sage: (Frob^2).order()
6
sage: (Frob^9).order()
4
"""
if self._order == 0:
from sage.rings.infinity import Infinity
return Infinity
else:
return Integer(self._order)
def power(self):
"""
Return an integer `n` such that this endomorphism
is the `n`-th power of the absolute (arithmetic)
Frobenius.
EXAMPLES::
sage: k.<t> = GF(5^12)
sage: Frob = k.frobenius_endomorphism()
sage: Frob.power()
1
sage: (Frob^9).power()
9
sage: (Frob^13).power()
1
"""
return self._power
def __pow__(self, n, modulus):
"""
Return the `n`-th iterate of this endomorphism.
EXAMPLES::
sage: k.<t> = GF(5^12)
sage: Frob = k.frobenius_endomorphism(); Frob
Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^12
sage: Frob^2
Frobenius endomorphism t |--> t^(5^2) on Finite Field in t of size 5^12
The result is simplified if possible::
sage: Frob^15
Frobenius endomorphism t |--> t^(5^3) on Finite Field in t of size 5^12
sage: Frob^36
Identity endomorphism of Finite Field in t of size 5^12
"""
return self.__class__(self.domain(), self.power()*n)
def _composition(self, right):
"""
Return self o right.
EXAMPLES::
sage: k.<t> = GF(5^12)
sage: f = k.frobenius_endomorphism(); f
Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^12
sage: g = k.frobenius_endomorphism(2); g
Frobenius endomorphism t |--> t^(5^2) on Finite Field in t of size 5^12
sage: f * g
Frobenius endomorphism t |--> t^(5^3) on Finite Field in t of size 5^12
The result is simplified if possible::
sage: f = k.frobenius_endomorphism(9)
sage: g = k.frobenius_endomorphism(10)
sage: f * g
Frobenius endomorphism t |--> t^(5^7) on Finite Field in t of size 5^12
"""
if isinstance(right, FrobeniusEndomorphism_finite_field):
return self.__class__(self.domain(), self._power + right.power())
else:
return RingHomomorphism._composition(self, right)
def fixed_field(self):
"""
Return the fixed field of ``self``.
OUTPUT:
- a tuple `(K, e)`, where `K` is the subfield of the domain
consisting of elements fixed by ``self`` and `e` is an
embedding of `K` into the domain.
.. NOTE::
The name of the variable used for the subfield (if it
is not a prime subfield) is suffixed by ``_fixed``.
EXAMPLES::
sage: k.<t> = GF(5^6)
sage: f = k.frobenius_endomorphism(2)
sage: kfixed, embed = f.fixed_field()
sage: kfixed
Finite Field in t_fixed of size 5^2
sage: embed
Ring morphism:
From: Finite Field in t_fixed of size 5^2
To: Finite Field in t of size 5^6
Defn: t_fixed |--> 4*t^5 + 2*t^4 + 4*t^2 + t
sage: tfixed = kfixed.gen()
sage: embed(tfixed)
4*t^5 + 2*t^4 + 4*t^2 + t
"""
if self._degree_fixed == 1:
k = FiniteField(self.domain().characteristic())
from .hom_prime_finite_field import FiniteFieldHomomorphism_prime
f = FiniteFieldHomomorphism_prime(Hom(k, self.domain()))
else:
k = FiniteField(self.domain().characteristic()**self._degree_fixed,
name=self.domain().variable_name() + "_fixed")
f = FiniteFieldHomomorphism_generic(Hom(k, self.domain()))
return k, f
def is_injective(self):
"""
Return true since any power of the Frobenius endomorphism
over a finite field is always injective.
EXAMPLES::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: Frob.is_injective()
True
"""
return True
def is_surjective(self):
"""
Return true since any power of the Frobenius endomorphism
over a finite field is always surjective.
EXAMPLES::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: Frob.is_surjective()
True
"""
return True
def is_identity(self):
"""
Return true if this morphism is the identity morphism.
EXAMPLES::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: Frob.is_identity()
False
sage: (Frob^3).is_identity()
True
"""
return self.power() == 0
def __hash__(self):
r"""
Return a hash of this morphism
EXAMPLES::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: hash(Frob) # random
383183030479672104
"""
return Morphism.__hash__(self)
cdef dict _extra_slots(self):
r"""
Helper function for copying and pickling
TESTS::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism(2)
sage: Frob.__reduce__() # indirect doctest
(<built-in function unpickle_map>,
(<class 'sage.rings.finite_rings.hom_finite_field_givaro.FrobeniusEndomorphism_givaro'>,
Automorphism group of Finite Field in t of size 5^3,
{},
{'_codomain': Finite Field in t of size 5^3,
'_domain': Finite Field in t of size 5^3,
'_is_coercion': False,
'_lift': None,
'_power': 2,
'_repr_type_str': None}))
"""
cdef dict slots
slots = FrobeniusEndomorphism_generic._extra_slots(self)
slots['_power'] = self._power
return slots
cdef _update_slots(self, dict slots):
r"""
Helper function for copying and pickling
TESTS::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism(2)
sage: Frob
Frobenius endomorphism t |--> t^(5^2) on Finite Field in t of size 5^3
sage: phi = copy(Frob)
sage: phi
Frobenius endomorphism t |--> t^(5^2) on Finite Field in t of size 5^3
sage: Frob == phi
True
"""
FrobeniusEndomorphism_generic._update_slots(self, slots)
self._power = slots['_power']
domain = self.domain()
self._degree = domain.degree()
self._degree_fixed = domain.degree().gcd(self._power)
self._order = self._degree / self._degree_fixed
self._q = domain.characteristic() ** self._power
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.rings.finite_field_morphism', 'FiniteFieldHomomorphism_generic', FiniteFieldHomomorphism_generic)