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)