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

class sage.categories.complete_discrete_valuation.CompleteDiscreteValuationFields[source]

Bases: Category_singleton

The category of complete discrete valuation fields.

EXAMPLES:

sage: Zp(7) in CompleteDiscreteValuationFields()                                # needs sage.rings.padics
False
sage: QQ in CompleteDiscreteValuationFields()
False
sage: LaurentSeriesRing(QQ, 'u') in CompleteDiscreteValuationFields()
True
sage: Qp(7) in CompleteDiscreteValuationFields()                                # needs sage.rings.padics
True
sage: TestSuite(CompleteDiscreteValuationFields()).run()

>>> from sage.all import *
>>> Zp(Integer(7)) in CompleteDiscreteValuationFields()                                # needs sage.rings.padics
False
>>> QQ in CompleteDiscreteValuationFields()
False
>>> LaurentSeriesRing(QQ, 'u') in CompleteDiscreteValuationFields()
True
>>> Qp(Integer(7)) in CompleteDiscreteValuationFields()                                # needs sage.rings.padics
True
>>> TestSuite(CompleteDiscreteValuationFields()).run()
class ElementMethods[source]

Bases: object

denominator()[source]

Return the denominator of this element normalized as a power of the uniformizer

EXAMPLES:

sage: # needs sage.rings.padics
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)

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> K = Qp(Integer(7))
>>> x = K(Integer(1)/Integer(21))
>>> x.denominator()
7 + O(7^21)
>>> x = K(Integer(7))
>>> x.denominator()
1 + O(7^20)

Note that the denominator lives in the ring of integers:

sage: x.denominator().parent()                                          # needs sage.rings.padics
7-adic Ring with capped relative precision 20

>>> from sage.all import *
>>> x.denominator().parent()                                          # needs sage.rings.padics
7-adic Ring with capped relative precision 20

When the denominator is indistinguishable from 0 and the precision on the input is \(O(p^n)\), the return value is \(1\) if \(n\) is nonnegative and \(p^(-n)\) otherwise:

sage: # needs sage.rings.padics
sage: x = K(0, 5); x
O(7^5)
sage: x.denominator()
1 + O(7^20)
sage: x = K(0, -5); x
O(7^-5)
sage: x.denominator()
7^5 + O(7^25)

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> x = K(Integer(0), Integer(5)); x
O(7^5)
>>> x.denominator()
1 + O(7^20)
>>> x = K(Integer(0), -Integer(5)); x
O(7^-5)
>>> x.denominator()
7^5 + O(7^25)
numerator()[source]

Return the numerator of this element, normalized in such a way that \(x = x.numerator() / x.denominator()\) always holds true.

EXAMPLES:

sage: # needs sage.rings.padics
sage: K = Qp(7, 5)
sage: x = K(1/21)
sage: x.numerator()
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + O(7^5)
sage: x == x.numerator() / x.denominator()
True

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> K = Qp(Integer(7), Integer(5))
>>> x = K(Integer(1)/Integer(21))
>>> x.numerator()
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + O(7^5)
>>> x == x.numerator() / x.denominator()
True

Note that the numerator lives in the ring of integers:

sage: x.numerator().parent()                                            # needs sage.rings.padics
7-adic Ring with capped relative precision 5

>>> from sage.all import *
>>> x.numerator().parent()                                            # needs sage.rings.padics
7-adic Ring with capped relative precision 5
valuation()[source]

Return the valuation of this element.

EXAMPLES:

sage: K = Qp(7)                                                         # needs sage.rings.padics
sage: x = K(7); x                                                       # needs sage.rings.padics
7 + O(7^21)
sage: x.valuation()                                                     # needs sage.rings.padics
1

>>> from sage.all import *
>>> K = Qp(Integer(7))                                                         # needs sage.rings.padics
>>> x = K(Integer(7)); x                                                       # needs sage.rings.padics
7 + O(7^21)
>>> x.valuation()                                                     # needs sage.rings.padics
1
super_categories()[source]

EXAMPLES:

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

>>> from sage.all import *
>>> CompleteDiscreteValuationFields().super_categories()
[Category of discrete valuation fields]
class sage.categories.complete_discrete_valuation.CompleteDiscreteValuationRings[source]

Bases: Category_singleton

The category of complete discrete valuation rings.

EXAMPLES:

sage: Zp(7) in CompleteDiscreteValuationRings()                                 # needs sage.rings.padics
True
sage: QQ in CompleteDiscreteValuationRings()
False
sage: QQ[['u']] in CompleteDiscreteValuationRings()
True
sage: Qp(7) in CompleteDiscreteValuationRings()                                 # needs sage.rings.padics
False
sage: TestSuite(CompleteDiscreteValuationRings()).run()

>>> from sage.all import *
>>> Zp(Integer(7)) in CompleteDiscreteValuationRings()                                 # needs sage.rings.padics
True
>>> QQ in CompleteDiscreteValuationRings()
False
>>> QQ[['u']] in CompleteDiscreteValuationRings()
True
>>> Qp(Integer(7)) in CompleteDiscreteValuationRings()                                 # needs sage.rings.padics
False
>>> TestSuite(CompleteDiscreteValuationRings()).run()
class ElementMethods[source]

Bases: object

denominator()[source]

Return the denominator of this element normalized as a power of the uniformizer

EXAMPLES:

sage: # needs sage.rings.padics
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)

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> K = Qp(Integer(7))
>>> x = K(Integer(1)/Integer(21))
>>> x.denominator()
7 + O(7^21)
>>> x = K(Integer(7))
>>> x.denominator()
1 + O(7^20)

Note that the denominator lives in the ring of integers:

sage: x.denominator().parent()                                          # needs sage.rings.padics
7-adic Ring with capped relative precision 20

>>> from sage.all import *
>>> x.denominator().parent()                                          # needs sage.rings.padics
7-adic Ring with capped relative precision 20

When the denominator is indistinguishable from 0 and the precision on the input is \(O(p^n)\), the return value is \(1\) if \(n\) is nonnegative and \(p^(-n)\) otherwise:

sage: # needs sage.rings.padics
sage: x = K(0, 5); x
O(7^5)
sage: x.denominator()
1 + O(7^20)
sage: x = K(0, -5); x
O(7^-5)
sage: x.denominator()
7^5 + O(7^25)

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> x = K(Integer(0), Integer(5)); x
O(7^5)
>>> x.denominator()
1 + O(7^20)
>>> x = K(Integer(0), -Integer(5)); x
O(7^-5)
>>> x.denominator()
7^5 + O(7^25)
lift_to_precision(absprec=None)[source]

Return another element of the same parent with absolute precision at least absprec, congruent to this element modulo the precision of this element.

INPUT:

  • absprec – integer or None (default: None); the absolute precision of the result. If None, lifts to the maximum precision allowed.

Note

If setting absprec that high would violate the precision cap, raises a precision error. Note that the new digits will not necessarily be zero.

EXAMPLES:

sage: # needs sage.rings.padics
sage: R = ZpCA(17)
sage: R(-1, 2).lift_to_precision(10)
16 + 16*17 + O(17^10)
sage: R(1, 15).lift_to_precision(10)
1 + O(17^15)
sage: R(1, 15).lift_to_precision(30)
Traceback (most recent call last):
...
PrecisionError: precision higher than allowed by the precision cap
sage: (R(-1, 2).lift_to_precision().precision_absolute()
....:   == R.precision_cap())
True

sage: R = Zp(5); c = R(17, 3); c.lift_to_precision(8)                   # needs sage.rings.padics
2 + 3*5 + O(5^8)
sage: c.lift_to_precision().precision_relative() == R.precision_cap()   # needs sage.rings.padics
True

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> R = ZpCA(Integer(17))
>>> R(-Integer(1), Integer(2)).lift_to_precision(Integer(10))
16 + 16*17 + O(17^10)
>>> R(Integer(1), Integer(15)).lift_to_precision(Integer(10))
1 + O(17^15)
>>> R(Integer(1), Integer(15)).lift_to_precision(Integer(30))
Traceback (most recent call last):
...
PrecisionError: precision higher than allowed by the precision cap
>>> (R(-Integer(1), Integer(2)).lift_to_precision().precision_absolute()
...   == R.precision_cap())
True

>>> R = Zp(Integer(5)); c = R(Integer(17), Integer(3)); c.lift_to_precision(Integer(8))                   # needs sage.rings.padics
2 + 3*5 + O(5^8)
>>> c.lift_to_precision().precision_relative() == R.precision_cap()   # needs sage.rings.padics
True
numerator()[source]

Return the numerator of this element, normalized in such a way that \(x = x.numerator() / x.denominator()\) always holds true.

EXAMPLES:

sage: # needs sage.rings.padics
sage: K = Qp(7, 5)
sage: x = K(1/21)
sage: x.numerator()
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + O(7^5)
sage: x == x.numerator() / x.denominator()
True

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> K = Qp(Integer(7), Integer(5))
>>> x = K(Integer(1)/Integer(21))
>>> x.numerator()
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + O(7^5)
>>> x == x.numerator() / x.denominator()
True

Note that the numerator lives in the ring of integers:

sage: x.numerator().parent()                                            # needs sage.rings.padics
7-adic Ring with capped relative precision 5

>>> from sage.all import *
>>> x.numerator().parent()                                            # needs sage.rings.padics
7-adic Ring with capped relative precision 5
valuation()[source]

Return the valuation of this element.

EXAMPLES:

sage: R = Zp(7)                                                         # needs sage.rings.padics
sage: x = R(7); x                                                       # needs sage.rings.padics
7 + O(7^21)
sage: x.valuation()                                                     # needs sage.rings.padics
1

>>> from sage.all import *
>>> R = Zp(Integer(7))                                                         # needs sage.rings.padics
>>> x = R(Integer(7)); x                                                       # needs sage.rings.padics
7 + O(7^21)
>>> x.valuation()                                                     # needs sage.rings.padics
1
super_categories()[source]

EXAMPLES:

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

>>> from sage.all import *
>>> CompleteDiscreteValuationRings().super_categories()
[Category of discrete valuation rings]