View source for P-adic number

An element of an extension of the field of rational numbers (cf.
[[Extension of a field|Extension of a field]]) based on the
divisibility of integers by a given prime number $p$. The extension is
obtained by completing the field of rational numbers with respect to a
non-Archimedean valuation (cf.
[[Norm on a field]]).

A $p$-adic integer, for an arbitrary prime number $p$, is a sequence
$x=(x_0,x_1,\dots)$ of residues $x_n$ modulo $p^{n+1}$ which satisfy the condition 
$$x_n\equiv x_{n-1} \mod p^n,\quad n\ge 1$$
The
addition and the multiplication of $p$-adic integers is defined by the
formulas 
$$(x+y)_n \equiv x_n+y_n \mod p^{n+1},$$

$$(xy)_n \equiv x_n y_n \mod p^{n+1},$$
Each integer $m$ is identified with the $p$-adic number
$x=(m,m,\dots)$. With respect to addition and multiplication, the $p$-adic
integers form a ring which contains the ring of integers. The ring of
$p$-adic integers may also be defined as the projective limit 
$$\def\plim#1{\lim_\underset{#1}{\longleftarrow}\;}\plim{n}\Z/p^n\Z$$
of
residues modulo $p^n$ (with respect to the natural projections).

A $p$-adic number, or rational $p$-adic number, is an element of the
quotient field $\Q_p$ of the ring $\Z_p$ of $p$-adic integers. This field is
called the field of $p$-adic numbers and it contains the field of
rational numbers as a subfield. Both the ring and the field of
$p$-adic numbers carry a natural topology. This topology may be
defined by a metric connected with the [[p-adic valuation|$p$-adic norm]], i.e. with the
function $|x|_p$ of the $p$-adic number $x$ which is defined as
follows. If $x\ne 0$, $x$ can be uniquely represented as $p^n a$, where $a$ is
an invertible element of the ring of $p$-adic integers. The $p$-adic
norm $|x|_p$ is then equal to $p^{-n}$. If $x=0$, then $|x|_p = 0$. If $|x|_p$ is initially
defined on rational numbers only, the field of $p$-adic numbers can be
obtained as the completion of the field of rational numbers with
respect to the $p$-adic norm.

Each element of the field of $p$-adic numbers may be represented in
the form 
$$x=\sum_{k=k_0}^\infty a_kp^k,\quad 0\le a_k <p,\label{*}$$
where $a_k$ are integers, $k_0$ is some integer, $a_{k_0}$, and
the series (*) converges in the metric of the field $\Q_p$. The numbers
$x\in\Q_p$ with $|x|_p\le 1$ (i.e. $k_0\ge 0$) form the ring $\Z_p$ of $p$-adic integers, which
is the completion of the ring of integers $\Z$ of the field $\Q$. The
numbers $x\in\Z_p$ with $|x|_p = 1$ (i.e. $k_0=0$, $a_0\ne 0$) form a multiplicative group and
are called $p$-adic units. The set of numbers $x\in\Z_p$ with $|x|_p < 1$ (i.e. $k_0\ge 1$)
forms a principal ideal in $\Z_p$ with generating element $p$. The ring
$\Z_p$ is a complete discrete valuation ring (cf. also
[[Discretely-normed ring|Discretely-normed ring]]). The field $\Q_p$ is
locally compact in the topology induced by the metric $|x-x'|_p$. It
therefore admits an invariant measure $\mu$, usually taken with the
condition $\mu(\Z_p) = 1$. For different $p$, the valuations $|x|_p$ are independent,
and the fields $\Q_p$ are non-isomorphic. Numerous facts and concepts of
classical analysis can be generalized to the case of $p$-adic fields.

$p$-adic numbers are connected with the solution of Diophantine
equations modulo increasing powers of a prime number. Thus, if $F(x_1,\dots,x_m)$ is
a polynomial with integral coefficients, the solvability, for all $k\ge 1$,
of the congruence 
$$F(x_1,\dots,x_m)\equiv 0 \mod p^k$$
is equivalent to the solvability of the
equation $F(x_1,\dots,x_m) = 0$ in $p$-adic integers. A necessary condition for the
solvability of this equation in integers or in rational numbers is its
solvability in the rings or, correspondingly, in the fields of
$p$-adic numbers for all $p$. Such an approach to the solution of
Diophantine equations and, in particular, the question whether these
conditions — the so-called local conditions — are sufficient,
constitutes an important branch of modern number theory (cf.
[[Diophantine geometry|Diophantine geometry]]).

The above solvability condition may in one special case be replaced by
a simpler one. In fact, if 
$$F(x_1,\dots,x_m)\equiv 0 \mod p$$
has a solution $({\bar x}_1,\dots,{\bar x}_m)$ and if this
solution defines a non-singular point of the hypersurface ${\bar F}(x_1,\dots,x_m) = 0 $, where
$\bar F$ is the polynomial $F$ modulo $p$, then this equation has a
solution in $p$-adic integers which is congruent to $({\bar x}_1,\dots,{\bar x}_m)$ modulo
$p$. This theorem, which is known as the
[[Hensel lemma|Hensel lemma]], is a special case of a more general
fact in the theory of schemes.

The ring of $p$-adic integers may be regarded as a special case of the
construction of Witt rings $W(A)$. The ring of $p$-adic integers is
obtained if $A=\F_p$ is the finite field of $p$ elements (cf.
[[Witt vector|Witt vector]]). Another generalization of $p$-adic
numbers are $\mathfrak{p}$-adic numbers, resulting from the completion of
algebraic number fields with respect to non-Archimedean valuations
connected with prime divisors.

$p$-adic numbers were introduced by K. Hensel
[[#References|[1]]]. Their canonical representation (*) is analogous
to the expansion of analytic functions into power series. This is one
of the manifestations of the analogy between algebraic numbers and
algebraic functions.

====References====
<table><TR><TD valign="top">[1]</TD>
 <TD valign="top"> K. Hensel, "Ueber eine neue Begründung der Theorie der algebraischen Zahlen" ''Jahresber. Deutsch. Math.-Verein'' , '''6''' : 1 (1899) pp. 83–88</TD>
</TR><TR><TD valign="top">[2]</TD>
 <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)</TD>
</TR><TR><TD valign="top">[3]</TD>
 <TD valign="top"> S. Lang, "Algebraic numbers" , Springer (1986)</TD>
</TR><TR><TD valign="top">[4]</TD>
 <TD valign="top"> H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959)</TD>
</TR><TR><TD valign="top">[5]</TD>
 <TD valign="top"> H. Hasse, "Zahlentheorie" , Akademie Verlag (1963)</TD>
</TR><TR><TD valign="top">[6]</TD>
 <TD valign="top"> A. Weil, "Basic number theory" , Springer (1974)</TD>
</TR><TR><TD valign="top">[7]</TD>
 <TD valign="top"> N. Bourbaki, "Elements of mathematics" , '''7. Commutative algebra''' , Addison-Wesley (1972) (Translated from French)</TD>
</TR></table>