Discrete Valuation Rings (DVR) and Fields (DVF)#
- class sage.categories.discrete_valuation.DiscreteValuationFields(s=None)#
Bases:
Category_singleton
The category of discrete valuation fields
EXAMPLES:
sage: Qp(7) in DiscreteValuationFields() True sage: TestSuite(DiscreteValuationFields()).run()
- class ElementMethods#
Bases:
object
- valuation()#
Return the valuation of this element.
EXAMPLES:
sage: x = Qp(5)(50) sage: x.valuation() 2
- class ParentMethods#
Bases:
object
- residue_field()#
Return the residue field of the ring of integers of this discrete valuation field.
EXAMPLES:
sage: Qp(5).residue_field() Finite Field of size 5 sage: K.<u> = LaurentSeriesRing(QQ) sage: K.residue_field() Rational Field
- uniformizer()#
Return a uniformizer of this ring.
EXAMPLES:
sage: Qp(5).uniformizer() 5 + O(5^21)
- super_categories()#
EXAMPLES:
sage: DiscreteValuationFields().super_categories() [Category of fields]
- class sage.categories.discrete_valuation.DiscreteValuationRings(s=None)#
Bases:
Category_singleton
The category of discrete valuation rings
EXAMPLES:
sage: GF(7)[['x']] in DiscreteValuationRings() True sage: TestSuite(DiscreteValuationRings()).run()
- class ElementMethods#
Bases:
object
- euclidean_degree()#
Return the Euclidean degree of this element.
- gcd(other)#
Return the greatest common divisor of self and other, normalized so that it is a power of the distinguished uniformizer.
- is_unit()#
Return True if self is invertible.
EXAMPLES:
sage: x = Zp(5)(50) sage: x.is_unit() False sage: x = Zp(7)(50) sage: x.is_unit() True
- lcm(other)#
Return the least common multiple of self and other, normalized so that it is a power of the distinguished uniformizer.
- quo_rem(other)#
Return the quotient and remainder for Euclidean division of
self
byother
.
- valuation()#
Return the valuation of this element.
EXAMPLES:
sage: x = Zp(5)(50) sage: x.valuation() 2
- class ParentMethods#
Bases:
object
- residue_field()#
Return the residue field of this ring.
EXAMPLES:
sage: Zp(5).residue_field() Finite Field of size 5 sage: K.<u> = QQ[[]] sage: K.residue_field() Rational Field
- uniformizer()#
Return a uniformizer of this ring.
EXAMPLES:
sage: Zp(5).uniformizer() 5 + O(5^21) sage: K.<u> = QQ[[]] sage: K.uniformizer() u
- super_categories()#
EXAMPLES:
sage: DiscreteValuationRings().super_categories() [Category of euclidean domains]