from sage.rings.homset import RingHomset_generic
from sage.rings.morphism import RingHomomorphism_im_gens
from sage.rings.integer import Integer
from sage.structure.sequence import Sequence
class FiniteFieldHomset(RingHomset_generic):
"""
Set of homomorphisms with domain a given finite field.
"""
def __call__(self, im_gens, check=True):
"""
EXAMPLES::
sage: R.<t> = ZZ[]
sage: E.<a> = GF(25, modulus = t^2 - 2)
sage: F.<b> = GF(625)
sage: End(E)
Automorphism group of Finite Field in a of size 5^2
sage: list(Hom(E, F))
[Ring morphism:
From: Finite Field in a of size 5^2
To: Finite Field in b of size 5^4
Defn: a |--> 4*b^3 + 4*b^2 + 4*b,
Ring morphism:
From: Finite Field in a of size 5^2
To: Finite Field in b of size 5^4
Defn: a |--> b^3 + b^2 + b]
sage: [phi(2*a)^2 for phi in Hom(E, F)]
[3, 3]
sage: End(GF(7))[0]
Ring endomorphism of Finite Field of size 7
Defn: 1 |--> 1
sage: H = Hom(GF(7), GF(49, 'c'))
sage: H[0](2)
2
sage: Hom(GF(49, 'c'), GF(7)).list()
[]
sage: Hom(GF(49, 'c'), GF(81, 'd')).list()
[]
sage: H = Hom(GF(9, 'a'), GF(81, 'b'))
sage: H == loads(dumps(H))
True
"""
if isinstance(im_gens, FiniteFieldHomomorphism_im_gens):
return self._coerce_impl(im_gens)
try:
return FiniteFieldHomomorphism_im_gens(self, im_gens, check=check)
except (NotImplementedError, ValueError), err:
try:
return self._coerce_impl(im_gens)
except TypeError:
raise TypeError, "images do not define a valid homomorphism"
def _coerce_impl(self, x):
"""
Coercion of other morphisms.
EXAMPLES::
sage: k.<a> = GF(25)
sage: l.<b> = GF(625)
sage: H = Hom(k, l)
sage: G = loads(dumps(H))
sage: H == G
True
sage: H is G # this should change eventually
False
sage: G.coerce(list(H)[0]) # indirect doctest
Ring morphism:
From: Finite Field in a of size 5^2
To: Finite Field in b of size 5^4
Defn: a |--> 4*b^3 + 4*b^2 + 4*b + 3
"""
if not isinstance(x, FiniteFieldHomomorphism_im_gens):
raise TypeError
if x.parent() is self:
return x
if x.parent() == self:
return FiniteFieldHomomorphism_im_gens(self, x.im_gens())
raise TypeError
def _repr_(self):
"""
EXAMPLES::
sage: Hom(GF(4, 'a'), GF(16, 'b'))._repr_()
'Set of field embeddings from Finite Field in a of size 2^2 to Finite Field in b of size 2^4'
sage: Hom(GF(4, 'a'), GF(4, 'c'))._repr_()
'Set of field embeddings from Finite Field in a of size 2^2 to Finite Field in c of size 2^2'
sage: Hom(GF(4, 'a'), GF(4, 'a'))._repr_()
'Automorphism group of Finite Field in a of size 2^2'
"""
D = self.domain()
C = self.codomain()
if C == D:
return "Automorphism group of %s"%D
else:
return "Set of field embeddings from %s to %s"%(D, C)
def is_aut(self):
"""
EXAMPLES::
sage: Hom(GF(4, 'a'), GF(16, 'b')).is_aut()
False
sage: Hom(GF(4, 'a'), GF(4, 'c')).is_aut()
False
sage: Hom(GF(4, 'a'), GF(4, 'a')).is_aut()
True
"""
return self.domain() == self.codomain()
def order(self):
"""
Return the order of this set of field homomorphisms.
EXAMPLES::
sage: K.<a> = GF(125)
sage: End(K)
Automorphism group of Finite Field in a of size 5^3
sage: End(K).order()
3
sage: L.<b> = GF(25)
sage: Hom(L, K).order() == Hom(K, L).order() == 0
True
"""
try:
return self.__order
except AttributeError:
pass
n = len(self.list())
self.__order = n
return n
def list(self):
"""
Return a list of all the elements in this set of field homomorphisms.
EXAMPLES::
sage: K.<a> = GF(25)
sage: End(K)
Automorphism group of Finite Field in a of size 5^2
sage: list(End(K))
[Ring endomorphism of Finite Field in a of size 5^2
Defn: a |--> 4*a + 1,
Ring endomorphism of Finite Field in a of size 5^2
Defn: a |--> a]
sage: L.<z> = GF(7^6)
sage: [g for g in End(L) if (g^3)(z) == z]
[Ring endomorphism of Finite Field in z of size 7^6
Defn: z |--> 5*z^4 + 5*z^3 + 4*z^2 + 3*z + 1,
Ring endomorphism of Finite Field in z of size 7^6
Defn: z |--> 3*z^5 + 5*z^4 + 5*z^2 + 2*z + 3,
Ring endomorphism of Finite Field in z of size 7^6
Defn: z |--> z]
TESTS:
Check that Trac #11390 is fixed::
sage: K = GF(1<<16,'a'); L = GF(1<<32,'b')
sage: K.Hom(L)[0]
Ring morphism:
From: Finite Field in a of size 2^16
To: Finite Field in b of size 2^32
Defn: a |--> b^29 + b^27 + b^26 + b^23 + b^21 + b^19 + b^18 + b^16 + b^14 + b^13 + b^11 + b^10 + b^9 + b^8 + b^7 + b^6 + b^5 + b^2 + b
"""
try:
return self.__list
except AttributeError:
pass
D = self.domain()
C = self.codomain()
if D.characteristic() == C.characteristic() and Integer(D.degree()).divides(Integer(C.degree())):
f = D.modulus()
g = C['x'](f)
r = g.roots()
v = [self(D.hom(a, C)) for a, _ in r]
v = Sequence(v, immutable=True, cr=True)
else:
v = Sequence([], immutable=True, cr=False)
self.__list = v
return v
def __getitem__(self, n):
"""
EXAMPLES::
sage: H = Hom(GF(32, 'a'), GF(1024, 'b'))
sage: H[1]
Ring morphism:
From: Finite Field in a of size 2^5
To: Finite Field in b of size 2^10
Defn: a |--> b^7 + b^5
sage: H[2:4]
[
Ring morphism:
From: Finite Field in a of size 2^5
To: Finite Field in b of size 2^10
Defn: a |--> b^8 + b^6 + b^2,
Ring morphism:
From: Finite Field in a of size 2^5
To: Finite Field in b of size 2^10
Defn: a |--> b^9 + b^7 + b^6 + b^5 + b^4
]
"""
return self.list()[n]
def index(self, item):
"""
EXAMPLES::
sage: K.<z> = GF(1024)
sage: g = End(K)[3]
sage: End(K).index(g) == 3
True
"""
return self.list().index(item)
class FiniteFieldHomomorphism_im_gens(RingHomomorphism_im_gens):
pass
from sage.structure.sage_object import register_unpickle_override
register_unpickle_override('sage.rings.finite_field_morphism', 'FiniteFieldHomset', FiniteFieldHomset)
register_unpickle_override('sage.rings.finite_field_morphism', 'FiniteFieldHomomorphism_im_gens', FiniteFieldHomomorphism_im_gens)
