def complex_conjugation(self, P=None):
"""
Return the unique element of self corresponding to complex conjugation,
for a specified embedding P into the complex numbers. If P is not
specified, use the "standard" embedding, whenever that is well-defined.
EXAMPLES::
sage: L.<z> = CyclotomicField(7)
sage: G = L.galois_group()
sage: conj = G.complex_conjugation(); conj
(1,4)(2,5)(3,6)
sage: conj(z)
-z^5 - z^4 - z^3 - z^2 - z - 1
An example where the field is not CM, so complex conjugation really
depends on the choice of embedding::
sage: L = NumberField(x^6 + 40*x^3 + 1372,'a')
sage: G = L.galois_group()
sage: [G.complex_conjugation(x) for x in L.places()]
[(1,3)(2,6)(4,5), (1,5)(2,4)(3,6), (1,2)(3,4)(5,6)]
"""
if P is None:
Q = self.number_field().specified_complex_embedding()
if Q is None:
raise ValueError("No default complex embedding specified")
P = Q
P = refine_embedding(P, infinity)
if not self.number_field().is_galois():
raise TypeError("Extension is not Galois")
if self.number_field().is_totally_real():
raise TypeError("No complex conjugation (field is real)")
g = self.number_field().gen()
gconj = P(g).conjugate()
elts = [s for s in self if P(s(g)) == gconj]
if len(elts) != 1:
raise ArithmeticError("Something has gone very wrong here")
return elts[0]
def complex_conjugation(self, P=None):
"""
Return the unique element of self corresponding to complex conjugation,
for a specified embedding P into the complex numbers. If P is not
specified, use the "standard" embedding, whenever that is well-defined.
EXAMPLES::
sage: L.<z> = CyclotomicField(7)
sage: G = L.galois_group()
sage: conj = G.complex_conjugation(); conj
(1,4)(2,5)(3,6)
sage: conj(z)
-z^5 - z^4 - z^3 - z^2 - z - 1
An example where the field is not CM, so complex conjugation really
depends on the choice of embedding::
sage: L = NumberField(x^6 + 40*x^3 + 1372,'a')
sage: G = L.galois_group()
sage: [G.complex_conjugation(x) for x in L.places()]
[(1,3)(2,6)(4,5), (1,5)(2,4)(3,6), (1,2)(3,4)(5,6)]
"""
if P is None:
Q = self.number_field().specified_complex_embedding()
if Q is None:
raise ValueError("No default complex embedding specified")
P = Q
P = refine_embedding(P, infinity)
if not self.number_field().is_galois():
raise TypeError("Extension is not Galois")
if self.number_field().is_totally_real():
raise TypeError("No complex conjugation (field is real)")
g = self.number_field().gen()
gconj = P(g).conjugate()
elts = [s for s in self if P(s(g)) == gconj]
if len(elts) != 1:
raise ArithmeticError("Something has gone very wrong here")
return elts[0]
def complex_conjugation(self, P=None):
"""
Return the unique element of self corresponding to complex conjugation,
for a specified embedding P into the complex numbers. If P is not
specified, use the "standard" embedding, whenever that is well-defined.
EXAMPLE::
sage: L = CyclotomicField(7)
sage: G = L.galois_group()
sage: G.complex_conjugation()
(1,6)(2,3)(4,5)
An example where the field is not CM, so complex conjugation really
depends on the choice of embedding::
sage: L = NumberField(x^6 + 40*x^3 + 1372,'a')
sage: G = L.galois_group()
sage: [G.complex_conjugation(x) for x in L.places()]
[(1,3)(2,6)(4,5), (1,5)(2,4)(3,6), (1,2)(3,4)(5,6)]
"""
if P is None:
Q = self.splitting_field().specified_complex_embedding()
if Q is None:
raise ValueError, "No default complex embedding specified"
P = Q
from sage.rings.real_mpfr import is_RealField
from sage.rings.qqbar import is_AlgebraicRealField
if is_RealField(P.codomain()) or is_AlgebraicRealField(P.codomain()):
return self.one()
P = refine_embedding(P, infinity)
if self._non_galois_not_impl:
raise TypeError, "Extension is not Galois"
g = self.splitting_field().gen()
gconj = P(g).conjugate()
elts = [s for s in self if P(s(g)) == gconj]
if len(elts) != 1: raise ArithmeticError, "Something has gone very wrong here"
return elts[0]
