In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number p rather than ten, and extending to the left rather than to the right.

For example, comparing the expansion of the rational number ${\displaystyle {\tfrac {1}{5}}}$ in base 3 vs. the 3-adic expansion,

${\displaystyle {\begin{alignedat}{3}{\tfrac {1}{5}}&{}=0.01210121\ldots \ ({\text{base }}3)&&{}=0\cdot 3^{0}+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\{\tfrac {1}{5}}&{}=\dots 121012102\ \ ({\text{3-adic}})&&{}=\cdots +2\cdot 3^{3}+1\cdot 3^{2}+0\cdot 3^{1}+2\cdot 3^{0}.\end{alignedat}}}$

Formally, given a prime number p, a p-adic number can be defined as a series

${\displaystyle s=\sum _{i=k}^{\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\cdots }$

where k is an integer (possibly negative), and each ${\displaystyle a_{i}}$ is an integer such that ${\displaystyle 0\leq a_{i}<p.}$ A p-adic integer is a p-adic number such that ${\displaystyle k\geq 0.}$

In general the series that represents a p-adic number is not convergent in the usual sense, but it is convergent for the p-adic absolute value ${\displaystyle |s|_{p}=p^{-k},}$ where k is the least integer i such that ${\displaystyle a_{i}\neq 0}$ (if all ${\displaystyle a_{i}}$ are zero, one has the zero p-adic number, which has 0 as its p-adic absolute value).

Every rational number can be uniquely expressed as the sum of a series as above, with respect to the p-adic absolute value. This allows considering rational numbers as special p-adic numbers, and alternatively defining the p-adic numbers as the completion of the rational numbers for the p-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.

p-adic numbers were first described by Kurt Hensel in 1897,^[1] though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using p-adic numbers.^{[note 1]}

Motivation

Roughly speaking, modular arithmetic modulo a positive integer n consists of "approximating" every integer by the remainder of its division by n, called its residue modulo n. The main property of modular arithmetic is that the residue modulo n of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo n. If one knows that the absolute value of the result is less than n/2, this allows a computation of the result which does not involve any integer larger than n.

For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying the Chinese remainder theorem for recovering the result modulo the product of the moduli.

Another method discovered by Kurt Hensel consists of using a prime modulus p, and applying Hensel's lemma for recovering iteratively the result modulo ${\displaystyle p^{2},p^{3},\ldots ,p^{n},\ldots }$ If the process is continued infinitely, this provides eventually a result which is a p-adic number.

Basic lemmas

The theory of p-adic numbers is fundamentally based on the two following lemmas

Every nonzero rational number can be written ${\textstyle p^{v}{\frac {m}{n}},}$ where v, m, and n are integers and neither m nor n is divisible by p. The exponent v is uniquely determined by the rational number and is called its p-adic valuation (this definition is a particular case of a more general definition, given below). The proof of the lemma results directly from the fundamental theorem of arithmetic.

Every nonzero rational number r of valuation v can be uniquely written ${\displaystyle r=ap^{v}+s,}$ where s is a rational number of valuation greater than v, and a is an integer such that ${\displaystyle 0<a<p.}$

The proof of this lemma results from modular arithmetic: By the above lemma, ${\textstyle r=p^{v}{\frac {m}{n}},}$ where m and n are integers coprime with p. The modular inverse of n is an integer q such that ${\displaystyle nq=1+ph}$ for some integer h. Therefore, one has ${\textstyle {\frac {1}{n}}=q-p{\frac {h}{n}},}$Zdroj:https://en.wikipedia.org?pojem=P-adic_numbers
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