Complete Discrete Valuation Rings (CDVR) and Fields (CDVF)#
- class sage.categories.complete_discrete_valuation.CompleteDiscreteValuationFields(s=None)#
Bases:
Category_singletonThe category of complete discrete valuation fields
EXAMPLES:
sage: Zp(7) in CompleteDiscreteValuationFields() # optional - sage.rings.padics False sage: QQ in CompleteDiscreteValuationFields() False sage: LaurentSeriesRing(QQ, 'u') in CompleteDiscreteValuationFields() True sage: Qp(7) in CompleteDiscreteValuationFields() # optional - sage.rings.padics True sage: TestSuite(CompleteDiscreteValuationFields()).run()
- class ElementMethods#
Bases:
object- denominator()#
Return the denominator of this element normalized as a power of the uniformizer
EXAMPLES:
sage: K = Qp(7) # optional - sage.rings.padics sage: x = K(1/21) # optional - sage.rings.padics sage: x.denominator() # optional - sage.rings.padics 7 + O(7^21) sage: x = K(7) # optional - sage.rings.padics sage: x.denominator() # optional - sage.rings.padics 1 + O(7^20)
Note that the denominator lives in the ring of integers:
sage: x.denominator().parent() # optional - 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: x = K(0, 5); x # optional - sage.rings.padics O(7^5) sage: x.denominator() # optional - sage.rings.padics 1 + O(7^20) sage: x = K(0, -5); x # optional - sage.rings.padics O(7^-5) sage: x.denominator() # optional - sage.rings.padics 7^5 + O(7^25)
- numerator()#
Return the numerator of this element, normalized in such a way that \(x = x.numerator() / x.denominator()\) always holds true.
EXAMPLES:
sage: K = Qp(7, 5) # optional - sage.rings.padics sage: x = K(1/21) # optional - sage.rings.padics sage: x.numerator() # optional - sage.rings.padics 5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + O(7^5) sage: x == x.numerator() / x.denominator() # optional - sage.rings.padics True
Note that the numerator lives in the ring of integers:
sage: x.numerator().parent() # optional - sage.rings.padics 7-adic Ring with capped relative precision 5
- valuation()#
Return the valuation of this element.
EXAMPLES:
sage: K = Qp(7) # optional - sage.rings.padics sage: x = K(7); x # optional - sage.rings.padics 7 + O(7^21) sage: x.valuation() # optional - sage.rings.padics 1
- super_categories()#
EXAMPLES:
sage: CompleteDiscreteValuationFields().super_categories() [Category of discrete valuation fields]
- class sage.categories.complete_discrete_valuation.CompleteDiscreteValuationRings(s=None)#
Bases:
Category_singletonThe category of complete discrete valuation rings
EXAMPLES:
sage: Zp(7) in CompleteDiscreteValuationRings() # optional - sage.rings.padics True sage: QQ in CompleteDiscreteValuationRings() False sage: QQ[['u']] in CompleteDiscreteValuationRings() True sage: Qp(7) in CompleteDiscreteValuationRings() # optional - sage.rings.padics False sage: TestSuite(CompleteDiscreteValuationRings()).run()
- class ElementMethods#
Bases:
object- denominator()#
Return the denominator of this element normalized as a power of the uniformizer
EXAMPLES:
sage: K = Qp(7) # optional - sage.rings.padics sage: x = K(1/21) # optional - sage.rings.padics sage: x.denominator() # optional - sage.rings.padics 7 + O(7^21) sage: x = K(7) # optional - sage.rings.padics sage: x.denominator() # optional - sage.rings.padics 1 + O(7^20)
Note that the denominator lives in the ring of integers:
sage: x.denominator().parent() # optional - 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: x = K(0, 5); x # optional - sage.rings.padics O(7^5) sage: x.denominator() # optional - sage.rings.padics 1 + O(7^20) sage: x = K(0, -5); x # optional - sage.rings.padics O(7^-5) sage: x.denominator() # optional - sage.rings.padics 7^5 + O(7^25)
- lift_to_precision(absprec=None)#
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– an integer orNone(default:None), the absolute precision of the result. IfNone, lifts to the maximum precision allowed.
Note
If setting
absprecthat high would violate the precision cap, raises a precision error. Note that the new digits will not necessarily be zero.EXAMPLES:
sage: R = ZpCA(17) # optional - sage.rings.padics sage: R(-1, 2).lift_to_precision(10) # optional - sage.rings.padics 16 + 16*17 + O(17^10) sage: R(1, 15).lift_to_precision(10) # optional - sage.rings.padics 1 + O(17^15) sage: R(1, 15).lift_to_precision(30) # optional - sage.rings.padics Traceback (most recent call last): ... PrecisionError: precision higher than allowed by the precision cap sage: (R(-1, 2).lift_to_precision().precision_absolute() # optional - sage.rings.padics ....: == R.precision_cap()) True sage: R = Zp(5); c = R(17, 3); c.lift_to_precision(8) # optional - sage.rings.padics 2 + 3*5 + O(5^8) sage: c.lift_to_precision().precision_relative() == R.precision_cap() # optional - sage.rings.padics True
- numerator()#
Return the numerator of this element, normalized in such a way that \(x = x.numerator() / x.denominator()\) always holds true.
EXAMPLES:
sage: K = Qp(7, 5) # optional - sage.rings.padics sage: x = K(1/21) # optional - sage.rings.padics sage: x.numerator() # optional - sage.rings.padics 5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + O(7^5) sage: x == x.numerator() / x.denominator() # optional - sage.rings.padics True
Note that the numerator lives in the ring of integers:
sage: x.numerator().parent() # optional - sage.rings.padics 7-adic Ring with capped relative precision 5
- valuation()#
Return the valuation of this element.
EXAMPLES:
sage: R = Zp(7) # optional - sage.rings.padics sage: x = R(7); x # optional - sage.rings.padics 7 + O(7^21) sage: x.valuation() # optional - sage.rings.padics 1
- super_categories()#
EXAMPLES:
sage: CompleteDiscreteValuationRings().super_categories() [Category of discrete valuation rings]