P-adic number
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.
References
[1] | K. Hensel, "Ueber eine neue Begründung der Theorie der algebraischen Zahlen" Jahresber. Deutsch. Math.-Verein , 6 : 1 (1899) pp. 83–88 |
[2] | Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) |
[3] | S. Lang, "Algebraic numbers" , Springer (1986) |
[4] | H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959) |
[5] | H. Hasse, "Zahlentheorie" , Akademie Verlag (1963) |
[6] | A. Weil, "Basic number theory" , Springer (1974) |
[7] | N. Bourbaki, "Elements of mathematics" , 7. Commutative algebra , Addison-Wesley (1972) (Translated from French) |
P-adic number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-adic_number&oldid=16260