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