Padic -- p-adic numbers

Description

The field of $p$-adic numbers $\QQ_p$ consists of all formal Laurent series $$\sum_{n=\nu}^\infty a_n p^n = a_\nu p^\nu + a_{\nu+1} p^{\nu+1} + \cdots,$$ where $\nu \in \ZZ$ and $a_n \in \{0, \ldots, p - 1\}$, together with the usual operations of addition and multiplication in base $p$, with carrying that may continue indefinitely. Equivalently, $\QQ_p$ is the completion of $\QQ$ with respect to the $p$-adic absolute value $|x|_p = p^{-\nu_p(x)}$, just as $\RR$ is the completion of $\QQ$ under the usual absolute value.

Elements of $\QQ_p$ are created by applying a p-adic field family to a rational number. They are stored and displayed as truncated series, with a default precision of 20 base-$p$ digits.

i1 : x = QQ_7(12/7)

o1 = 5*7^-1 + 1

o1 : QQ  (of precision 20)
       7

This package is implemented using the ForeignFunctions package to call $p$-adic arithmetic routines from the FLINT C library.

See the paper Implementing p-adic numbers in Macaulay2 using its foreign function interface and FLINT for more information.

Author

Doug Torrance <dtorrance@piedmont.edu>

Version

This documentation describes version 0.1 of Padic, released April 28, 2026.

Citation

If you have used this package in your research, please cite it as follows:

@misc{torrance2026implementingpadicnumbersmacaulay2,
  title={Implementing p-adic numbers in Macaulay2 using its foreign function interface and FLINT},
  author={Douglas A. Torrance},
  year={2026},
  eprint={2604.16799},
  archivePrefix={arXiv},
  primaryClass={math.AG},
  url={https://arxiv.org/abs/2604.16799},
}

Exports

Types
- PadicFieldFamily -- class for p-adic fields
- PadicNumber -- base class for p-adic numbers
Functions and commands
- prime -- prime of a p-adic field or number
- pVal -- p-adic valuation of a p-adic number
- teichmullerLift -- Teichmüller lift of a p-adic integer
- unit -- unit part of a p-adic number
Methods
- 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
- Number ^ PadicFieldFamily -- see 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
- PadicFieldFamily Thing (missing documentation)
- 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
- padicValuation(PadicFieldFamily) (missing documentation)
- precision(PadicNumber) -- precision of a p-adic number
- prime(PadicFieldFamily) -- see prime -- prime of a p-adic field or number
- prime(PadicNumber) -- see prime -- prime of a p-adic field or number
- Number _ PadicFieldFamily -- see promote(ZZ,type of PadicNumber) -- promote a number to a p-adic field
- 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
- valuation(PadicFieldFamily) (missing documentation)

For the programmer

The object Padic is a package, defined in Padic.m2.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/Padic.m2:439:0.