P-adic number

In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers that 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

1

5

{\displaystyle {\tfrac {1}{5}}}

in base 3 versus the 3-adic expansion,

1

5

=

0.

0121

¯

(

base

3

)

=

0

⋅

3

0

+

0

⋅

3

−

1

+

1

⋅

3

−

2

+

2

⋅

3

−

3

+

⋯

1

5

=

1210

¯

2

(

3-adic

)

=

⋯

+

2

⋅

3

3

+

1

⋅

3

2

+

0

⋅

3

1

+

2

⋅

3

0

.

{\displaystyle {\begin{alignedat}{3}{\tfrac {1}{5}}&{}=0.{\overline {0121}}\ ({\text{base }}3)&&{}=0\cdot 3^{0}+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\[5mu]{\tfrac {1}{5}}&{}={\overline {1210}}2\ \ ({\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

s

=

∑

i

=

k

∞

a

i

p

i

=

a

k

p

k

+

a

k

+

1

p

k

+

1

+

a

k

+

2

p

k

+

2

+

⋯

{\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

a

i

{\displaystyle a_{i}}

is an integer such that

0

≤

a

i

<

p

.

{\displaystyle 0\leq a_{i}<p.}

A p-adic integer is a p-adic number such that

k

≥

0.

{\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

|

s

|

p

=

p

−

k

,

{\displaystyle |s|_{p}=p^{-k},}

where k is the least integer i such that

a

i

≠

0

{\displaystyle a_{i}\neq 0}

(if all

a

i

{\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, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using p-adic numbers.

For every prime number p, the p-adic numbers form a field.