P-adic valuation explained

In number theory, the valuation or -adic order of an integer is the exponent of the highest power of the prime number that divides .It is denoted

\nup(n)

.Equivalently,

\nup(n)

is the exponent to which

p

appears in the prime factorization of

n

.

The -adic valuation is a valuation and gives rise to an analogue of the usual absolute value.Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers

R

, the completion of the rational numbers with respect to the

p

-adic absolute value results in the numbers

Qp

.[1]

Definition and properties

Let be a prime number.

Integers

The -adic valuation of an integer

n

is defined to be

\nup(n)= \begin{cases} max\{k\inN0:pk\midn\}&ifn0\\ infty&ifn=0, \end{cases}

where

N0

denotes the set of natural numbers (including zero) and

m\midn

denotes divisibility of

n

by

m

. In particular,

\nup

is a function

\nup\colonZ\toN0\cup\{infty\}

.[2]

For example,

\nu2(-12)=2

,

\nu3(-12)=1

, and

\nu5(-12)=0

since

|{-12}|=12=223150

.

The notation

pk\paralleln

is sometimes used to mean

k=\nup(n)

.[3]

If

n

is a positive integer, then

\nup(n)\leqlogpn

this follows directly from

n\geq

\nup(n)
p
.

Rational numbers

The -adic valuation can be extended to the rational numbers as the function

\nup:Q\toZ\cup\{infty\}

[4] [5]

defined by

\nu
p\left(r
s

\right)=\nup(r)-\nup(s).

For example,

\nu2l(\tfrac{9}{8}r)=-3

and

\nu3l(\tfrac{9}{8}r)=2

since

\tfrac{9}{8}=2-332

.

Some properties are:

\nup(rs)=\nup(r)+\nup(s)

\nup(r+s)\geqminl\{\nup(r),\nup(s)r\}

Moreover, if

\nup(r)\nup(s)

, then

\nup(r+s)=minl\{\nup(r),\nup(s)r\}

where

min

is the minimum (i.e. the smaller of the two).

-adic absolute value

The -adic absolute value (or -adic norm,[6] though not a norm in the sense of analysis) on

Q

is the function

||p\colon\Q\to\R\ge

defined by

|r|p=

-\nup(r)
p

.

Thereby,

|0|p=p-infty=0

for all

p

and for example,

|{-12}|2=2-2=\tfrac{1}{4}

and

l|\tfrac{9}{8}r|2=2-(-3)=8.

The -adic absolute value satisfies the following properties.

Non-negativity
r_p \geq 0
r_p = 0 \iff r = 0
r s_p = r_ps_p
r+s_p \leq \max\left(r_p, s_p\right)
From the multiplicativity

|rs|p=|r|p|s|p

it follows that

|1|p=1=|-1|p

for the roots of unity

1

and

-1

and consequently also

|{-r}|p=|r|p.

The subadditivity

|r+s|p\leq|r|p+|s|p

follows from the non-Archimedean triangle inequality

|r+s|p\leqmax\left(|r|p,|s|p\right)

.
-\nup(r)
p

makes no difference for most of the properties, but supports the product formula:

\prod0,|r|p=1

where the product is taken over all primes and the usual absolute value, denoted

|r|0

. This follows from simply taking the prime factorization: each prime power factor

pk

contributes its reciprocal to its -adic absolute value, and then the usual Archimedean absolute value cancels all of them.

A metric space can be formed on the set

Q

with a (non-Archimedean, translation-invariant) metric

d\colon\Q x \Q\to\R\ge

defined by

d(r,s)=|r-s|p.

The completion of

Q

with respect to this metric leads to the set

Qp

of -adic numbers.

See also

Notes and References

  1. Book: David S.. Dummit . Richard M. . Foote . 2003 . Abstract Algebra . 3rd . Wiley . 0-471-43334-9 . 758–759.
  2. Book: Ireland . K. . Rosen . M. . 2000 . A Classical Introduction to Modern Number Theory . Springer-Verlag . New York . 3.
  3. Book: Niven . Ivan . Ivan M. Niven . Zuckerman . Herbert S. . Montgomery . Hugh L. . Hugh Lowell Montgomery . An Introduction to the Theory of Numbers . 1991 . . 5th . 0-471-62546-9 . 4.
  4. with the usual order relation, namely

    infty>n

    ,and rules for arithmetic operations,

    infty+n=n+infty=infty

    ,on the extended number line.
  5. Book: Khrennikov . A. . Nilsson . M. . 2004 . -adic Deterministic and Random Dynamics . Kluwer Academic Publishers . 9.
  6. Book: Murty, M. Ram . 10.1007/978-1-4757-3441-6 . 0-387-95143-1 . 1803093 . 147–148 . Springer-Verlag, New York . Graduate Texts in Mathematics . Problems in analytic number theory . 206 . 2001.