PadicNumber -- base class for p-adic numbers

Description

Every $p$-adic number is an instance of the class QQ_p for the appropriate value of p. However, each of of these classes is a subclass of PadicNumber.

i1 : x = QQ_7 5

o1 = 5

o1 : QQ  (of precision 20)
       7

i2 : ancestors class x

o2 = {QQ , PadicNumber, Number, Thing}
        7

o2 : List

To install methods that work for all $p$-adic numbers, regardless of the specific field, install methods for this class.

i3 : foo = method()

o3 = foo

o3 : MethodFunction

i4 : foo PadicNumber := x -> x + 2

o4 = FunctionClosure[currentString:4:19-4:29]

o4 : FunctionClosure

i5 : foo x

o5 = 1*7^1

o5 : QQ  (of precision 20)
       7

precision(PadicNumber) -- precision of a p-adic number
unit -- unit part of a p-adic number
pVal -- p-adic valuation of a p-adic number
PadicNumber == PadicNumber -- equality of p-adic numbers
PadicNumber + PadicNumber -- add p-adic numbers
PadicNumber * PadicNumber -- multiply p-adic numbers
abs(PadicNumber) -- p-adic absolute value
PadicNumber << ZZ -- multiply a p-adic number by a power of p
PadicNumber ^ ZZ -- raise a p-adic number to an integer power
exp(PadicNumber) -- p-adic exponential function
teichmullerLift -- Teichmüller lift of a p-adic integer
lift(PadicNumber,type of ZZ) -- lift a p-adic number to another ring
promote(ZZ,type of PadicNumber) -- promote a number to a p-adic field
numeric(PadicNumber) -- see numeric(ZZ,PadicNumber) -- convert a p-adic number to a real number
interval(PadicNumber) -- see interval(PadicNumber,PadicNumber) -- convert p-adic numbers to a real interval

Methods that use a p-adic number:

abs(PadicNumber) -- p-adic absolute value
exp(PadicNumber) -- p-adic exponential function
interval(Number,PadicNumber) -- see interval(PadicNumber,PadicNumber) -- convert p-adic numbers to a real interval
interval(PadicNumber) -- see interval(PadicNumber,PadicNumber) -- convert p-adic numbers to a real interval
interval(PadicNumber,Number) -- see interval(PadicNumber,PadicNumber) -- convert p-adic numbers to a real interval
interval(PadicNumber,PadicNumber) -- convert p-adic numbers to a real interval
inverse(PadicNumber) -- multiplicative inverse of a p-adic number
lift(PadicNumber,type of PadicNumber) -- see lift(PadicNumber,type of ZZ) -- lift a p-adic number to another ring
lift(PadicNumber,type of QQ) -- see lift(PadicNumber,type of ZZ) -- lift a p-adic number to another ring
lift(PadicNumber,type of ZZ) -- lift a p-adic number to another ring
log(PadicNumber) -- p-adic logarithm function
new PadicNumber from Constant (missing documentation)
new PadicNumber from Number (missing documentation)
new PadicNumber from PadicNumber (missing documentation)
NewFromMethod(PadicNumber,ZZ,Constant) (missing documentation)
NewFromMethod(PadicNumber,ZZ,Number) (missing documentation)
numeric(PadicNumber) -- see numeric(ZZ,PadicNumber) -- convert a p-adic number to a real number
numeric(ZZ,PadicNumber) -- convert a p-adic number to a real number
Number * PadicNumber -- see PadicNumber * PadicNumber -- multiply p-adic numbers
PadicNumber * Number -- see PadicNumber * PadicNumber -- multiply p-adic numbers
PadicNumber * PadicNumber -- multiply p-adic numbers
+ PadicNumber -- see PadicNumber + PadicNumber -- add p-adic numbers
Number + PadicNumber -- see PadicNumber + PadicNumber -- add p-adic numbers
PadicNumber + Number -- see PadicNumber + PadicNumber -- add p-adic numbers
PadicNumber + PadicNumber -- add p-adic numbers
- PadicNumber -- see PadicNumber - PadicNumber -- subtract or negate p-adic numbers
Number - PadicNumber -- see PadicNumber - PadicNumber -- subtract or negate p-adic numbers
PadicNumber - Number -- see PadicNumber - PadicNumber -- subtract or negate p-adic numbers
PadicNumber - PadicNumber -- subtract or negate p-adic numbers
Number / PadicNumber -- see PadicNumber / PadicNumber -- divide p-adic numbers
PadicNumber / Number -- see PadicNumber / PadicNumber -- divide p-adic numbers
PadicNumber / PadicNumber -- divide p-adic numbers
PadicNumber << ZZ -- multiply a p-adic number by a power of p
Number == PadicNumber -- see PadicNumber == PadicNumber -- equality of p-adic numbers
PadicNumber == Number -- see PadicNumber == PadicNumber -- equality of p-adic numbers
PadicNumber == PadicNumber -- equality of p-adic numbers
PadicNumber >> ZZ -- divide a p-adic number by a power of p
PadicNumber ^ ZZ -- raise a p-adic number to an integer power
precision(PadicNumber) -- precision of a p-adic number
prime(PadicNumber) -- see prime -- prime of a p-adic field or number
promote(PadicNumber,type of PadicNumber) -- see promote(ZZ,type of PadicNumber) -- promote a number to a p-adic field
promote(QQ,type of PadicNumber) -- see promote(ZZ,type of PadicNumber) -- promote a number to a p-adic field
promote(ZZ,type of PadicNumber) -- promote a number to a p-adic field
pVal(PadicNumber) -- see pVal -- p-adic valuation of a p-adic number
sqrt(PadicNumber) -- square root of a p-adic number
teichmullerLift(PadicNumber) -- see teichmullerLift -- Teichmüller lift of a p-adic integer
unit(PadicNumber) -- see unit -- unit part of a p-adic number

For the programmer

The object PadicNumber is a type, with ancestor classes Number < Thing.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/Padic.m2:508:0.

PadicNumber -- base class for p-adic numbers

Description

Menu

Methods that use a p-adic number:

For the programmer