Complete Discrete Valuation Rings (CDVR) and Fields (CDVF)¶
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class
sage.categories.complete_discrete_valuation.
CompleteDiscreteValuationFields
(s=None)¶ Bases:
sage.categories.category_singleton.Category_singleton
The category of complete discrete valuation fields
EXAMPLES:
sage: Zp(7) in CompleteDiscreteValuationFields() False sage: QQ in CompleteDiscreteValuationFields() False sage: LaurentSeriesRing(QQ,'u') in CompleteDiscreteValuationFields() True sage: Qp(7) in CompleteDiscreteValuationFields() True sage: TestSuite(CompleteDiscreteValuationFields()).run()
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class
ElementMethods
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denominator
()¶ Return the denominator of this element normalized as a power of the uniformizer
EXAMPLES:
sage: K = Qp(7) sage: x = K(1/21) sage: x.denominator() 7 + O(7^21) sage: x = K(7) sage: x.denominator() 1 + O(7^20)
Note that the denominator lives in the ring of integers:
sage: x.denominator().parent() 7-adic Ring with capped relative precision 20
An error is raised when the input is indistinguishable from 0:
sage: x = K(0,5); x O(7^5) sage: x.denominator() Traceback (most recent call last): ... ValueError: Cannot determine the denominator of an element indistinguishable from 0
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valuation
()¶ Return the valuation of this element.
EXAMPLES:
sage: K = Qp(7) sage: x = K(7); x 7 + O(7^21) sage: x.valuation() 1
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super_categories
()¶ EXAMPLES:
sage: CompleteDiscreteValuationFields().super_categories() [Category of discrete valuation fields]
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class
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class
sage.categories.complete_discrete_valuation.
CompleteDiscreteValuationRings
(s=None)¶ Bases:
sage.categories.category_singleton.Category_singleton
The category of complete discrete valuation rings
EXAMPLES:
sage: Zp(7) in CompleteDiscreteValuationRings() True sage: QQ in CompleteDiscreteValuationRings() False sage: QQ[['u']] in CompleteDiscreteValuationRings() True sage: Qp(7) in CompleteDiscreteValuationRings() False sage: TestSuite(CompleteDiscreteValuationRings()).run()
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class
ElementMethods
¶ -
denominator
()¶ Return the denominator of this element normalized as a power of the uniformizer
EXAMPLES:
sage: K = Qp(7) sage: x = K(1/21) sage: x.denominator() 7 + O(7^21) sage: x = K(7) sage: x.denominator() 1 + O(7^20)
Note that the denominator lives in the ring of integers:
sage: x.denominator().parent() 7-adic Ring with capped relative precision 20
An error is raised when the input is indistinguishable from 0:
sage: x = K(0,5); x O(7^5) sage: x.denominator() Traceback (most recent call last): ... ValueError: Cannot determine the denominator of an element indistinguishable from 0
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valuation
()¶ Return the valuation of this element.
EXAMPLES:
sage: R = Zp(7) sage: x = R(7); x 7 + O(7^21) sage: x.valuation() 1
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super_categories
()¶ EXAMPLES:
sage: CompleteDiscreteValuationRings().super_categories() [Category of discrete valuation rings]
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class