Complete Discrete Valuation Rings (CDVR) and Fields (CDVF)

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()
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
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
super_categories()

EXAMPLES:

sage: CompleteDiscreteValuationFields().super_categories()
[Category of discrete valuation fields]
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()
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
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
super_categories()

EXAMPLES:

sage: CompleteDiscreteValuationRings().super_categories()
[Category of discrete valuation rings]