An element of an extension of the field of rational numbers (cf.
Extension of a field)
based on the divisibility of integers by a given prime number
.
The extension is obtained by completing the field of
rational numbers with respect to a non-Archimedean valuation (cf.
Norm on a field).
A
-adic integer,
for an arbitrary prime number
,
is a sequence
of residues
modulo
which satisfy the condition
The addition and the multiplication of
-adic
integers is defined by the formulas
Each integer
is identified with the
-adic
number
.
With respect to addition and multiplication, the
-adic
integers form a ring which contains the ring of integers. The ring of
-adic
integers may also be defined as the projective limit
of residues modulo
(with respect to the natural projections).
A
-adic number,
or
rational
-adic number,
is an element of the quotient field
of the ring
of
-adic
integers. This field is called the
field of
-adic numbers
and it contains the field of rational numbers as
a subfield. Both the ring and the field of
-adic
numbers carry a natural topology. This topology may
be defined by a metric connected with the
-adic norm,
i.e. with the function
of the
-adic
number
which is defined as follows. If
,
can be uniquely represented as
,
where
is an invertible element of the ring of
-adic
integers. The
-adic
norm
is then equal to
.
If
,
then
.
If
is initially defined on rational numbers only, the field of
-adic
numbers can be obtained as the completion of the
field of rational numbers with respect to the
-adic
norm.
Each element of the field of
-adic
numbers may be represented in the form
where
are integers,
is some integer,
,
and the series
(*)
converges in the metric of the field
.
The numbers
with
(i.e.
)
form the ring
of
-adic
integers, which is the completion of the ring of integers
of the field
.
The numbers
with
(i.e.
,
)
form a multiplicative group and are called
-adic units.
The set of numbers
with
(i.e.
)
forms a principal ideal in
with generating element
.
The ring
is a complete discrete valuation ring (cf. also
Discretely-normed ring).
The field
is locally compact in the topology induced by the metric
.
It therefore admits an invariant measure
,
usually taken with the condition
.
For different
,
the valuations
are independent, and the fields
are non-isomorphic. Numerous facts and concepts of classical
analysis can be generalized to the case of
-adic
fields.
-adic
numbers are connected with the solution of Diophantine equations
modulo increasing powers of a prime number. Thus, if
is a polynomial with integral coefficients, the solvability, for all
,
of the congruence
is equivalent to the solvability of the equation
in
-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
-adic
numbers for all
.
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).
The above solvability condition may in one special case
be replaced by a simpler one. In fact, if
has a solution
and if this solution defines a non-singular point of the hypersurface
,
where
is the polynomial
modulo
,
then this equation has a solution in
-adic
integers which is congruent to
modulo
.
This theorem, which is known as the
Hensel lemma,
is a special case of a more general fact in the theory of schemes.
The ring of
-adic
integers may be regarded as a special case of the construction of Witt rings
.
The ring of
-adic
integers is obtained if
is the finite field of
elements (cf.
Witt vector).
Another generalization of
-adic
numbers are
-adic
numbers, resulting from the completion of algebraic number fields
with respect to non-Archimedean valuations connected with prime divisors.
-adic
numbers were introduced by
K. Hensel
[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.