Discrete Valuation Rings (DVR) and Fields (DVF)#

class sage.categories.discrete_valuation.DiscreteValuationFields#

Bases: Category_singleton

The category of discrete valuation fields

EXAMPLES:

sage: Qp(7) in DiscreteValuationFields()                                        # needs sage.rings.padics
True
sage: TestSuite(DiscreteValuationFields()).run()

class ElementMethods#

Bases: object

valuation()#

Return the valuation of this element.

EXAMPLES:

sage: # needs sage.rings.padics
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()                                             # needs sage.rings.padics
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()                                               # needs sage.rings.padics
5 + O(5^21)

super_categories()#

EXAMPLES:

sage: DiscreteValuationFields().super_categories()
[Category of fields]

class sage.categories.discrete_valuation.DiscreteValuationRings#

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: # needs sage.rings.padics
sage: x = Zp(5)(50)
sage: x.is_unit()
False

sage: # needs sage.rings.padics
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 by other.

EXAMPLES:

sage: R.<q> = GF(5)[[]]
sage: (q^2 + q).quo_rem(q)
(1 + q, 0)
sage: (q + 1).quo_rem(q^2)
(0, 1 + q)

valuation()#

Return the valuation of this element.

EXAMPLES:

sage: # needs sage.rings.padics
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()                                             # needs sage.rings.padics
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()                                               # needs sage.rings.padics
5 + O(5^21)

sage: K.<u> = QQ[[]]
sage: K.uniformizer()
u

super_categories()#

EXAMPLES:

sage: DiscreteValuationRings().super_categories()
[Category of euclidean domains]