valuation

valuation

Let

K

be a field. A

valuation

or

absolute value

on

K

is a

function

|

⋅

|

:

K

→

ℝ

satisfying the properties:

1.

|

x

|

≥

0

for all

x

∈

K

, with

equality

if and only if

x

=

0

2.

|

x

⁢

y

|

=

|

x

|

⋅

|

y

|

for all

x

,

y

∈

K

3.

|

x

+

y

|

≤

|

x

|

+

|

y

|

If a valuation

satisfies

|

x

+

y

|

≤

max

⁡

(

|

x

|

,

|

y

|

)

, then we say that it is a

non-archimedean valuation

. Otherwise we say that it is an

archimedean valuation

.

Every valuation on

K

defines a metric on

K

, given by

d

⁢

(

x

,

y

)

:=

|

x

-

y

|

. This metric is an ultrametric if and only if the valuation is non-archimedean. Two valuations are

equivalent

if their corresponding metrics induce the same topology on

K

. An

equivalence class

v

of valuations on

K

is called a

prime

of

K

. If

v

consists of archimedean valuations, we say that

v

is an

infinite prime

, or

archimedean

prime

. Otherwise, we say that

v

is a

finite prime

, or

non-archimedean prime

.

In the case where

K

is a

number field

, primes as defined above generalize the notion of

prime ideals

in the following way. Let

𝔭

⊂

K

be a nonzero prime ideal

1

1

By “prime ideal” we mean “prime

fractional ideal

of

K

” or equivalently “prime ideal of the

ring of integers

of

K

”. We do not mean literally a prime ideal of the ring

K

, which would be the

zero ideal

.

, considered as a fractional ideal. For every nonzero

element

x

∈

K

, let

r

be the unique

integer

such that

x

∈

𝔭

r

but

x

∉

𝔭

r

+

1

. Define

|

x

|

𝔭

:=

{

1

/

N

⁢

(

𝔭

)

r

x

≠

0

,

0

x

=

0

,

where

N

⁢

(

𝔭

)

denotes the

absolute norm

of

𝔭

. Then

|

⋅

|

𝔭

is a non–archimedean valuation on

K

, and furthermore every non-archimedean valuation on

K

is equivalent to

|

⋅

|

𝔭

for some prime ideal

𝔭

. Hence, the prime ideals of

K

correspond bijectively with the finite primes of

K

, and it is in this sense that the notion of primes as valuations generalizes that of a prime ideal.

As for the archimedean valuations, when

K

is a number field every

embedding

of

K

into

ℝ

or

ℂ

yields a valuation of

K

by way of the standard absolute value on

ℝ

or

ℂ

, and one can show that every archimedean valuation of

K

is equivalent to one arising in this way. Thus the infinite primes of

K

correspond to embeddings of

K

into

ℝ

or

ℂ

. Such a prime is called real or

complex

according to whether the valuations comprising it arise from real or

complex embeddings

.

Title

valuation

Canonical name

Valuation

Date of creation

2013-03-22 12:35:07

Last modified on

2013-03-22 12:35:07

Owner

djao (24)

Last modified by

djao (24)

Numerical id

17

Author

djao (24)

Entry type

Definition

Classification

msc 13F30

Classification

msc 13A18

Classification

msc 12J20

Classification

msc 11R99

Synonym

absolute value

Related topic

DiscreteValuationRing

Related topic

DiscreteValuation

Related topic

Ultrametric

Related topic

HenselianField

Defines

infinite prime

Defines

finite prime

Defines

archimedean

Defines

non-archimedean

Defines

real prime

Defines

complex prime

Defines

prime

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