order valuation

order valuation

Given a

Krull valuation

|

.

|

of a field

K

as a mapping of

K

to an

ordered group

G

(with operation “

⋅

”) equipped with

0

, one may use for the an alternative notation “ord”:

The “

<

” of

G

is reversed and the operation of

G

is denoted by “

+

”.  The element

0

of

G

is denoted as

∞

, thus

∞

is greater than any other element of

G

.  When we still call the

valuation

the

order

of

K

and instead of

|

x

|

write

ord

⁢

x

, the valuation postulates read as follows.

1.

ord

⁢

x

=

∞

iff

x

=

0

;

2.

ord

⁢

x

⁢

y

=

ord

⁢

x

+

ord

⁢

y

;

3.

ord

⁢

(

x

+

y

)

≧

min

⁡

{

ord

⁢

x

,

ord

⁢

y

}

.

We must emphasize that the

order valuation

is nothing else than a Krull valuation.  The name

order

comes from

complex analysis

, where the “places” zero (

http://planetmath.org/ZeroOfAFunction

) and pole (

http://planetmath.org/Pole

) of a

meromorphic function

with their orders have a fully analogous meaning as have the corresponding concepts place (

http://planetmath.org/PlaceOfField

) and order valuation in the

valuation theory

.  Thus also a place

φ

of a field is called a

zero

of an element

a

of the field, if

φ

⁢

(

a

)

=

0

,  and a

pole

of an element

b

of the field, if

φ

⁢

(

b

)

=

∞

;  then e.g. the

equation

φ

⁢

(

a

)

=

0

implies always that

ord

⁢

a

>

0

.

Example.

Let

p

be a given

positive

prime number

.  Any non-zero

rational number

x

can be uniquely expressed in the form

x

=

p

n

⁢

u

,

in which

n

is an

integer

and the rational number

u

is by

p

indivisible, i.e. when reduced to

lowest terms

,

p

divides neither its numerator nor its denominator.  If we define

ord

p

⁢

x

=

{

∞

⁢

for

⁢

x

=

0

,

n

⁢

for

⁢

x

=

p

n

⁢

u

≠

0

,

then the

function

ord

p

, defined in

ℚ

, clearly satisfies the above postulates of the order valuation.

In [2], an order valuation having only integer values is called the

exponent

of the field

(

der Exponent des Körpers

); this name apparently motivated by the exponent

n

of

p

.  Such an order valuation is a special case of the

discrete valuation

.  Note, that an arbitrary order valuation need not be a discrete valuation, since the values need not be

real numbers

.

References

1

E. Artin

:

Theory of

Algebraic Numbers

.  Lecture notes.  Mathematisches Institut, Göttingen (1959).

2

S. Borewicz & I. Safarevic

:

Zahlentheorie

.  Birkhäuser Verlag. Basel und Stuttgart (1966).

Title

order valuation

Canonical name

OrderValuation

Date of creation

2013-03-22 16:53:28

Last modified on

2013-03-22 16:53:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

19

Author

pahio (2872)

Entry type

Definition

Classification

msc 13F30

Classification

msc 13A18

Classification

msc 12J20

Classification

msc 11R99

Synonym

additive valuation

Related topic

KrullValuation

Related topic

Valuation

Related topic

PAdicValuation

Related topic

DiscreteValuation

Related topic

ZerosAndPolesOfRationalFunction

Related topic

AlternativeDefinitionOfValuation2

Related topic

StrictDivisibility

Related topic

ExponentValuation2

Related topic

DivisibilityOfPrimePowerBinomialCoefficients

Defines

exponent of field

Defines

zero

Defines

zero of an element

Defines

pole

Defines

pole of an element

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