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

\nu _{p(n)} .Equivalently, \nu _{p(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 Q _{p} .^[1]

Definition and properties

Let be a prime number.

Integers

The -adic valuation of an integer

n is defined to be\nu _{p(n)= \begin{cases} max\{k} \in N _{0} : p ^{k} \mid n\} & if n ≠ 0 \\ inf ty & if n =0 , \end{cases} where N _{0} denotes the set of natural numbers (including zero) and m \mid n denotes divisibility of n by m . In particular, \nu _{p} is a function \nu _{p} \colon Z \to N _{0} \cup\{ inf ty\} .^[2]

For example,

\nu _{2(-12)} = 2 , \nu _{3(-12)} = 1 , and \nu _{5(-12)} = 0 since |{ -12 }| = 12 = 2 ^{2} ⋅ 3 ^{1} ⋅ 5 ^{0} .

The notation

p ^{k} \parallel n is sometimes used to mean k = \nu _{p(n)} .^[3]

If

n is a positive integer, then\nu _{p(n)} \leq log _{p} n

this follows directly from

n \geq p ^{\nu _{p(n)}} .

Rational numbers

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

\nu _{p} : Q \to Z \cup\{ inf ty\} ^{[4] [5]}

defined by

\nu _{p\left( (r)/(s)} \right) = \nu _{p(r)-\nu} _{p(s).}

For example,

\nu _{2} l (\tfrac{ 9 }{ 8 } r ) = -3 and \nu _{3} l (\tfrac{ 9 }{ 8 } r ) = 2 since \tfrac{ 9 }{ 8 } = 2 ^{-3} ⋅ 3 ^{2} .

Some properties are:

\nu _{p(r ⋅} s) = \nu _{p(r)} + \nu _{p(s)} \nu _{p(r+s)} \geq min l \{ \nu _{p(r),} \nu _{p(s)r\}}

Moreover, if

\nu _{p(r)} ≠ \nu _{p(s)} , then\nu _{p(r+s)=} min l \{ \nu _{p(r),} \nu _{p(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} = p ^{- \nu _{p(r)}} .

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	_p	s	_p
		r+s	_p \leq \max\left(	r	_p,	s	_p\right)

From the multiplicativity |r s| _{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} \leq max \left(|r| _{p,} |s| _{p\right)} .p ^{- \nu _{p(r)}} makes no difference for most of the properties, but supports the product formula:\prod _{0,} |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 p ^{k} 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) metricd \colon \Q x \Q \to \R _{\ge} defined byd(r,s) = |r - s| _{p} . The completion of Q with respect to this metric leads to the set Q _{p} of -adic numbers.

See also

• -adic number
• Valuation (algebra)
• Archimedean property
• Multiplicity (mathematics)
• Ostrowski's theorem
• Legendre's formula

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, namelyinf ty > n ,and rules for arithmetic operations,inf ty + n = n + inf ty = inf ty ,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.