p-adic canonical form

p-adic canonical form

Every non-zero

p

-adic number (

p

is a

positive

rational prime

number) can be uniquely written in

canonical form

, formally as a Laurent series,

ξ

=

a

-

m

⁢

p

-

m

+

a

-

m

+

1

⁢

p

-

m

+

1

+

⋯

+

a

0

+

a

1

⁢

p

+

a

2

⁢

p

2

+

⋯

where

m

∈

ℕ

,

0

≦

a

k

≦

p

-

1

for all

k

’s, and at least one of the

integers

a

k

is positive.  In

addition

, we can write:

0

=

0

+

0

⁢

p

+

0

⁢

p

2

+

⋯

The field

ℚ

p

of the

p

-adic numbers is the

completion

of the field

ℚ

with respect to its

p

-adic

valuation

(

http://planetmath.org/PAdicValuation

); thus

ℚ

may be thought the

subfield

(

prime subfield

) of

ℚ

p

.  We can call the elements of

ℚ

p

∖

ℚ

the

proper

p

-adic numbers

.

If, e.g.,

p

=

2

,  we have the 2-

adic

or, according to G. W. Leibniz,

dyadic numbers

, for which every

a

k

is 0 or 1.  In this case we can write the sum expression for

ξ

in the reverse and use the ordinary positional (

http://planetmath.org/Base3

) (i.e.,

dyadic

) figure system (

http://planetmath.org/Base3

).  Then, for example, we have the

rational numbers

-

1

=

…

⁢

111111

,

1

=

…

⁢

0001

,

6.5

=

…

⁢

000110.1

,

1

5

=

…

⁢

00110011001101

.

(You may check the first by adding 1, and the last by multiplying by  5 = …000101.)
All 2-adic rational numbers have periodic binary expansion (

http://planetmath.org/DecimalExpansion

).  Similarly as the decimal (

http://planetmath.org/DecimalExpansion

) (according to Leibniz:

decadic

)

expansions

of

irrational

real numbers

are aperiodic, the proper 2-adic numbers also have aperiodic binary expansion, for example the 2-

adic

fractional number

α

=

…

⁢

1000010001001011.10111

.

The

2-adic fractional numbers

have some bits “1” after the

dyadic point

“.” (in continental Europe: comma “

,

”), the

2-adic integers

have none.  The 2-adic integers form a

subring

of the 2-adic field

ℚ

2

such that

ℚ

2

is the

quotient field

of this ring.

Every such 2-adic integer

ε

whose last bit is “1”, as

-

3

/

7

=

…

⁢

11011011011

, is a unit of this ring, because the

division

1

:

ε

clearly gives as quotient a integer (by the way, the divisions of the binary expansions in practice go from right to left and are very comfortable!).

Those integers ending in a “0” are non-units of the ring, and they apparently form the only

maximal ideal

in the ring (which thus is a

local ring

).  This is a

principal ideal

𝔭

, the

generator

of which may be taken

…

⁢

00010

=

10

(i.e., two).  Indeed, two is the only

prime number

of the ring, but it has infinitely many

associates

, a kind of copies, namely all expansions of the form

…

⁢

10

=

ε

⋅

10

.  The only non-trivial ideals in the ring of 2-adic integers are

𝔭

,

𝔭

2

,

𝔭

3

,

…

They have only 0 as common element.

All 2-adic non-zero integers are of the form

ε

⋅

2

n

where

n

=

0

,

1

,

2

,

…

.  The values

n

=

-

1

,

-

2

,

-

3

,

…

here would give non-integral, i.e. fractional 2-adic numbers.

If in the binary of an arbitrary 2-adic number, the last non-zero digit “1” corresponds to the power

2

n

, then the

2-adic valuation

of the 2-adic number

ξ

is given by

|

ξ

|

2

=

2

-

n

.

Title

p-adic canonical form

Canonical name

PadicCanonicalForm

Date of creation

2013-03-22 14:13:37

Last modified on

2013-03-22 14:13:37

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

66

Author

pahio (2872)

Entry type

Example

Classification

msc 12F99

Related topic

IntegralElement

Related topic

UltrametricTriangleInequality

Related topic

NonIsomorphicCompletionsOfMathbbQ

Related topic

IdealsOfADiscreteValuationRingArePowersOfItsMaximalIdeal

Defines

proper p-adic number

Defines

dyadic number

Defines

dyadic point

Defines

2-adic fractional number

Defines

2-adic integer

Defines

2-adic valuation

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