def valuation(self, p):
r"""
Return the discrete valuation with uniformizer ``p``.
EXAMPLES::
sage: v = QQ.valuation(3); v
3-adic valuation
sage: v(1/3)
-1
.. SEEALSO::
:meth:`NumberField_generic.valuation() <sage.rings.number_field.number_field.NumberField_generic.valuation>`,
:meth:`IntegerRing_class.valuation() <sage.rings.integer_ring.IntegerRing_class.valuation>`
"""
from sage.rings.padics.padic_valuation import pAdicValuation
return pAdicValuation(self, p)
def valuation(self, p):
r"""
Return the discrete valuation with uniformizer ``p``.
EXAMPLES::
sage: v = QQ.valuation(3); v
3-adic valuation
sage: v(1/3)
-1
.. SEEALSO::
:meth:`NumberField_generic.valuation() <sage.rings.number_field.number_field.NumberField_generic.valuation>`,
:meth:`IntegerRing_class.valuation() <sage.rings.integer_ring.IntegerRing_class.valuation>`
"""
from sage.rings.padics.padic_valuation import pAdicValuation
return pAdicValuation(self, p)
def valuation(self):
r"""
Return the `p`-adic valuation on this ring.
OUTPUT:
a valuation that is normalized such that the rational prime `p` has
valuation 1.
EXAMPLES::
sage: K = Qp(3)
sage: R.<a> = K[]
sage: L.<a> = K.extension(a^3 - 3)
sage: v = L.valuation(); v
3-adic valuation
sage: v(3)
1
sage: L(3).valuation()
3
The normalization is chosen such that the valuation restricts to the
valuation on the base ring::
sage: v(3) == K.valuation()(3)
True
sage: v.restriction(K) == K.valuation()
True
.. SEEALSO::
:meth:`NumberField_generic.valuation() <sage.rings.number_field.number_field.NumberField_generic.valuation>`,
:meth:`Order.valuation() <sage.rings.number_field.order.Order.valuation>`
"""
from sage.rings.padics.padic_valuation import pAdicValuation
return pAdicValuation(self)
def valuation(self):
r"""
Return the `p`-adic valuation on this ring.
OUTPUT:
a valuation that is normalized such that the rational prime `p` has
valuation 1.
EXAMPLES::
sage: K = Qp(3)
sage: R.<a> = K[]
sage: L.<a> = K.extension(a^3 - 3)
sage: v = L.valuation(); v
3-adic valuation
sage: v(3)
1
sage: L(3).valuation()
3
The normalization is chosen such that the valuation restricts to the
valuation on the base ring::
sage: v(3) == K.valuation()(3)
True
sage: v.restriction(K) == K.valuation()
True
.. SEEALSO::
:meth:`NumberField_generic.valuation() <sage.rings.number_field.number_field.NumberField_generic.valuation>`,
:meth:`Order.valuation() <sage.rings.number_field.order.Order.valuation>`
"""
from sage.rings.padics.padic_valuation import pAdicValuation
return pAdicValuation(self)