PAdic

The

PAdic

paclet adds the layer the Wolfram Language does not ship: for each prime

p

, a p-adic valuation

v

p

and its derived p-adic absolute value

|x

|

p



-

v

p

(x)

p

, the base-

p

digit expansion of any element of

Q

p

(extending

IntegerDigits

and

RealDigits

to elements with infinitely many digits to the left and finitely many to the right of the "decimal" point), and the Hensel lift that promotes a root of a polynomial mod

p

to a root mod

n

p

via Newton iteration. The valuation alone is enough to recover the ultrametric inequality

|x+y

|

p

≤max(|x

|

p

,|y

|

p

)

that makes

Q

p

a strikingly different completion of

Q

from the familiar

R

.

PAdicValuation

— the p-adic valuation

v

p

(x)

PAdicNorm

— the p-adic absolute value

|x

|

p



-

v

p

(x)

p

PAdicDigits

— the base-

p

digit expansion in

Q

p

(extends

RealDigits

to the p-adic completion)

HenselLift

— the p-adic Newton iteration that lifts

f(a)≡0(modp)

to

f(



a

)≡0(mod

n

p

)

PAdicNumber

— a Z_p element carrying

UpValues

for

Plus

,

Times

,

Power

,

Equal

, etc.

PAdicNumberQ

— predicate for normalised

PAdicNumber

expressions

PAdicDigitPlot

— bar chart of the first

n

base-

p

digits of

x

PAdicTree

— the Cayley tree of residues mod

k

p

- the ultrametric neighbourhood structure of

Z

p

PAdicValuationArray

— the Kummer / Sierpinski fractal

v

p

(

i+j

j

)

as an

ArrayPlot

Tech Notes

Related Guides

NumberTheoreticFunctions