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

class sage.categories.discrete_valuation.DiscreteValuationFields(s=None)

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)

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 by other.

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]