def field_extension(self, names):
r"""
Given a polynomial with base ring a quotient ring, return a
3-tuple: a number field defined by the same polynomial, a
homomorphism from its parent to the number field sending the
generators to one another, and the inverse isomorphism.
INPUT:
- ``names`` - name of generator of output field
OUTPUT:
- field
- homomorphism from self to field
- homomorphism from field to self
EXAMPLES::
sage: R.<x> = PolynomialRing(QQ)
sage: S.<alpha> = R.quotient(x^3-2)
sage: F.<a>, f, g = alpha.field_extension()
sage: F
Number Field in a with defining polynomial x^3 - 2
sage: a = F.gen()
sage: f(alpha)
a
sage: g(a)
alpha
Over a finite field, the corresponding field extension is not a
number field::
sage: R.<x> = GF(25,'b')['x']
sage: S.<a> = R.quo(x^3 + 2*x + 1)
sage: F.<b>, g, h = a.field_extension()
sage: h(b^2 + 3)
a^2 + 3
sage: g(x^2 + 2)
b^2 + 2
We do an example involving a relative number field::
sage: R.<x> = QQ['x']
sage: K.<a> = NumberField(x^3-2)
sage: S.<X> = K['X']
sage: Q.<b> = S.quo(X^3 + 2*X + 1)
sage: F, g, h = b.field_extension('c')
Another more awkward example::
sage: R.<x> = QQ['x']
sage: K.<a> = NumberField(x^3-2)
sage: S.<X> = K['X']
sage: f = (X+a)^3 + 2*(X+a) + 1
sage: f
X^3 + 3*a*X^2 + (3*a^2 + 2)*X + 2*a + 3
sage: Q.<z> = S.quo(f)
sage: F.<w>, g, h = z.field_extension()
sage: c = g(z)
sage: f(c)
0
sage: h(g(z))
z
sage: g(h(w))
w
AUTHORS:
- Craig Citro (2006-08-06)
- William Stein (2006-08-06)
"""
#TODO: is the return order backwards from the magma convention?
## We do another example over $\ZZ$.
## sage: R.<x> = ZZ['x']
## sage: S.<a> = R.quo(x^3 - 2)
## sage: F.<b>, g, h = a.field_extension()
## sage: h(b^2 + 3)
## a^2 + 3
## sage: g(x^2 + 2)
## a^2 + 2
## Note that the homomorphism is not defined on the entire
## ''domain''. (Allowing creation of such functions may be
## disallowed in a future version of Sage.): <----- INDEED!
## sage: h(1/3)
## Traceback (most recent call last):
## ...
## TypeError: Unable to coerce rational (=1/3) to an Integer.
## Note that the parent ring must be an integral domain:
## sage: R.<x> = GF(25,'b')['x']
## sage: S.<a> = R.quo(x^3 - 2)
## sage: F, g, h = a.field_extension()
## Traceback (most recent call last):
## ...
## ValueError: polynomial must be irreducible
R = self.parent()
x = R.gen()
F = R.modulus().root_field(names)
alpha = F.gen()
f = R.hom([alpha], F, check=False)
if number_field_rel.is_RelativeNumberField(F):
base_hom = F.base_field().hom([R.base_ring().gen()])
g = F.Hom(R)(x, base_hom)
else:
g = F.hom([x], R, check=False)
return F, f, g
def field_extension(self, names):
r"""
Given a polynomial with base ring a quotient ring, return a
3-tuple: a number field defined by the same polynomial, a
homomorphism from its parent to the number field sending the
generators to one another, and the inverse isomorphism.
INPUT:
- ``names`` - name of generator of output field
OUTPUT:
- field
- homomorphism from self to field
- homomorphism from field to self
EXAMPLES::
sage: R.<x> = PolynomialRing(QQ)
sage: S.<alpha> = R.quotient(x^3-2)
sage: F.<a>, f, g = alpha.field_extension()
sage: F
Number Field in a with defining polynomial x^3 - 2
sage: a = F.gen()
sage: f(alpha)
a
sage: g(a)
alpha
Over a finite field, the corresponding field extension is not a
number field::
sage: R.<x> = GF(25,'b')['x']
sage: S.<a> = R.quo(x^3 + 2*x + 1)
sage: F.<b>, g, h = a.field_extension()
sage: h(b^2 + 3)
a^2 + 3
sage: g(x^2 + 2)
b^2 + 2
We do an example involving a relative number field::
sage: R.<x> = QQ['x']
sage: K.<a> = NumberField(x^3-2)
sage: S.<X> = K['X']
sage: Q.<b> = S.quo(X^3 + 2*X + 1)
sage: F, g, h = b.field_extension('c')
Another more awkward example::
sage: R.<x> = QQ['x']
sage: K.<a> = NumberField(x^3-2)
sage: S.<X> = K['X']
sage: f = (X+a)^3 + 2*(X+a) + 1
sage: f
X^3 + 3*a*X^2 + (3*a^2 + 2)*X + 2*a + 3
sage: Q.<z> = S.quo(f)
sage: F.<w>, g, h = z.field_extension()
sage: c = g(z)
sage: f(c)
0
sage: h(g(z))
z
sage: g(h(w))
w
AUTHORS:
- Craig Citro (2006-08-06)
- William Stein (2006-08-06)
"""
#TODO: is the return order backwards from the magma convention?
## We do another example over $\ZZ$.
## sage: R.<x> = ZZ['x']
## sage: S.<a> = R.quo(x^3 - 2)
## sage: F.<b>, g, h = a.field_extension()
## sage: h(b^2 + 3)
## a^2 + 3
## sage: g(x^2 + 2)
## a^2 + 2
## Note that the homomorphism is not defined on the entire
## ''domain''. (Allowing creation of such functions may be
## disallowed in a future version of Sage.): <----- INDEED!
## sage: h(1/3)
## Traceback (most recent call last):
## ...
## TypeError: Unable to coerce rational (=1/3) to an Integer.
## Note that the parent ring must be an integral domain:
## sage: R.<x> = GF(25,'b')['x']
## sage: S.<a> = R.quo(x^3 - 2)
## sage: F, g, h = a.field_extension()
## Traceback (most recent call last):
## ...
## ValueError: polynomial must be irreducible
R = self.parent()
x = R.gen()
F = R.modulus().root_field(names)
alpha = F.gen()
f = R.hom([alpha], F, check=False)
if number_field_rel.is_RelativeNumberField(F):
base_hom = F.base_field().hom([R.base_ring().gen()])
g = F.Hom(R)(x, base_hom)
else:
g = F.hom([x], R, check=False)
return F, f, g