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p-adic integers∗
djao†
2013-03-21 14:24:29
1
Basic construction
For any prime p, the p–adic integers is the ring obtained by taking the completion of the integers Z with respect to the metric induced by the norm
|x| :=
1
, x ∈ Z,
pνp (x)
(1)
where νp (x) denotes the largest integer e such that pe divides x. The induced
metric d(x, y) := |x − y| is called the p–adic metric on Z. The ring of p–adic
integers is usually denoted by Zp , and its fraction field by Qp .
2
Profinite viewpoint
The ring Zp of p–adic integers can also be constructed by taking the inverse
limit
Zp := lim Z/pn Z
←−
over the inverse system · · · → Z/p2 Z → Z/pZ → 0 consisting of the rings
Z/pn Z, for all n ≥ 0, with the projection maps defined to be the unique maps
such that the diagram
Z EE
ww
EE
ww
EE
w
w
EE
w
E"
{ww
/ Z/pn Z
Z/pn+1 Z
commutes. An algebraic and topological isomorphism between the two constructions is obtained by taking the coordinatewise projection map Z → lim Z/pn Z,
←−
extended to the completion of Z under the p–adic metric.
∗ hPadicIntegersi
created: h2013-03-21i by: hdjaoi version: h33118i Privacy setting: h1i
hDefinitioni h11S99i h12J12i
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1
This alternate characterization shows that Zp is compact, since it is a closed
subspace of the space
Y
Z/pn Z
n≥0
which is an infinite product of finite topological spaces and hence compact under
the product topology.
3
Generalizations
If we interpret the prime p as an equivalence class of valuations on Q, then the
field Qp is simply the completion of the topological field Q with respect to the
metric induced by any member valuation of p (indeed, the valuation defined in
Equation (??), extended to Q, may serve as the representative). This notion
easily generalizes to other fields and valuations; namely, if K is any field, and
p is any prime of K, then the p–adic field Kp is defined to be the completion
of K with respect to any valuation in p. The analogue of the p–adic integers in
this case can be obtained by taking the subset (and subring) of Kp consisting
of all elements of absolute value less than or equal to 1, which is well defined
independent of the choice of valuation representing p.
In the special case where K is a number field, the p–adic ring Kp is always a
finite extension of Qp whenever p is a finite prime, and is always equal to either
R or C whenever p is an infinite prime.
2