lift(PadicNumber,type of ZZ) -- lift a p-adic number to another ring

Function: lift
Usage:

lift(x, R)

x^R
Inputs:
- x, a p-adic number,
- R, an integer, a rational number, or a p-adic number,
Optional inputs:
- Verify => ..., default value true
Outputs:
- a number, an element of R;

Description

Lift a $p$-adic number to an other ring. This is only well-defined when R to ZZ, QQ, or the $p$-adic field containing x.

i1 : x = QQ_7 49

o1 = 1*7^2

o1 : QQ  (of precision 20)
       7

i2 : lift(x, ZZ)

o2 = 49

i3 : lift(x, QQ)

o3 = 49

o3 : QQ

i4 : lift(x, QQ_7)

o4 = 1*7^2

o4 : QQ  (of precision 20)
       7

As usual, the ^ operator is a shorthand for this operation.

i5 : x^ZZ

o5 = 49

i6 : x^QQ

o6 = 49

o6 : QQ

i7 : x^(QQ_7)

o7 = 1*7^2

o7 : QQ  (of precision 20)
       7

Lifting to ZZ raises an error if the $p$-adic valuation of x is negative.

The source of this document is in Padic.m2:1028:0.