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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 $p$. 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 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 $$x=\sum_{k=k_0}^\infty a_kp^k,\quad 0\le a_k <p,\tag{1}$$ 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). 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).

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, 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). 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 [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)
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