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-adic integers

(Definition)

Basic construction

For any prime , the -adic integers is the ring obtained by taking the completion of the integers $\mathbb{Z}$ with respect to the metric induced by the norm

$\displaystyle \vert x\vert := \frac{1}{p^{\nu_p(x)}},\ \ x \in \mathbb{Z},$

(1)

where $\nu_p(x)$ denotes the largest integer

such that

divides

. The induced metric $d(x,y) := \vert x-y\vert$ is called the -adic metric on $\mathbb{Z}$ . The ring of

-adic integers is usually denoted by $\mathbb{Z}_p$ , and its fraction field by $\mathbb{Q}_p$ .

Profinite viewpoint

The ring $\mathbb{Z}_p$ of -adic integers can also be constructed by taking the inverse limit

$\displaystyle \mathbb{Z}_p := \,\underset{\longleftarrow}{\lim}\,\mathbb{Z}/p^n\mathbb{Z}$

over the inverse system $\cdots \to \mathbb{Z}/p^2\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}\to 0$ consisting of the rings $\mathbb{Z}/p^n\mathbb{Z}$ , for all $n \geq 0$ , with the projection maps defined to be the unique maps such that the diagram

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & \mathbb{Z}\ar[dl] \ar[dr]\ \mathbb{Z}/p^{n+1}\mathbb{Z}\ar[rr] & & \mathbb{Z}/p^n\mathbb{Z} } } \end{xy}$

commutes. An algebraic and topological isomorphism between the two constructions is obtained by taking the coordinatewise projection map $\mathbb{Z}\to \,\underset{\longleftarrow}{\lim}\,\mathbb{Z}/p^n\mathbb{Z}$ , extended to the completion of $\mathbb{Z}$ under the

-adic metric.

This alternate characterization shows that $\mathbb{Z}_p$ is compact, since it is a closed subspace of the space

$\displaystyle \prod_{n \geq 0} \mathbb{Z}/p^n\mathbb{Z}$

which is an infinite product of finite topological spaces and hence compact under the product topology.

Generalizations

If we interpret the prime as an equivalence class of valuations on $\mathbb{Q}$ , then the field $\mathbb{Q}_p$ is simply the completion of the topological field $\mathbb{Q}$ with respect to the metric induced by any member valuation of (indeed, the valuation defined in Equation (1), extended to $\mathbb{Q}$ , may serve as the representative). This notion easily generalizes to other fields and valuations; namely, if is any field, and $\mathfrak{p}$ is any prime of , then the $\mathfrak{p}$ -adic field $K_\mathfrak{p}$ is defined to be the completion of with respect to any valuation in $\mathfrak{p}$ . The analogue of the -adic integers in this case can be obtained by taking the subset (and subring) of $K_\mathfrak{p}$ consisting of all elements of absolute value less than or equal to , which is well defined independent of the choice of valuation representing $\mathfrak{p}$ .

In the special case where is a number field, the $\mathfrak{p}$ -adic ring $K_\mathfrak{p}$ is always a finite extension of $\mathbb{Q}_p$ whenever $\mathfrak{p}$ is a finite prime, and is always equal to either $\mathbb{R}$ or $\mathbb{C}$ whenever $\mathfrak{p}$ is an infinite prime.

-adic integers" is owned by djao.

(view preamble)

See Also: inverse limit

Other names:

-adic numbers, $\mathbb{Z}_p$

Attachments:: complex p-adic numbers (Definition) by alozano

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Cross-references: infinite prime, finite prime, finite extension, number field, independent, well defined, absolute value, subring, subset, equation, topological field, field, valuations, equivalence class, product topology, topological spaces, product, subspace, closed, compact, characterization, isomorphism, algebraic, maps, projection maps, inverse limit, fraction field, metric, induced, divides, metric induced by the norm, completion, ring, integers, prime
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This is version 7 of $p$

-adic integers, born on 2002-06-18, modified 2004-07-19.
Object id is 3118, canonical name is PAdicIntegers.
Accessed 10053 times total.

Classification:

AMS MSC:	11S99 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Miscellaneous)
	12J12 (Field theory and polynomials :: Topological fields :: Formally $p$-adic fields)

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