Euclidean valuation

Euclidean valuation

Let

D

be an

integral domain

. A

Euclidean valuation

is a

function

from the nonzero elements of

D

to the

nonnegative integers

ν

:

D

∖

{

0

D

}

→

{

x

∈

ℤ

:

x

≥

0

}

such that the following hold:

•

For any

a

,

b

∈

D

with

b

≠

0

D

, there exist

q

,

r

∈

D

such that

a

=

b

⁢

q

+

r

with

ν

⁢

(

r

)

<

ν

⁢

(

b

)

or

r

=

0

D

.

•

For any

a

,

b

∈

D

∖

{

0

D

}

, we have

ν

⁢

(

a

)

≤

ν

⁢

(

a

⁢

b

)

.

Euclidean valuations are important because they let us define

greatest common divisors

and use Euclid’s algorithm. Some facts about Euclidean valuations include:

•

The minimal (

http://planetmath.org/MinimalElement

) value of

ν

is

ν

⁢

(

1

D

)

. That is,

ν

⁢

(

1

D

)

≤

ν

⁢

(

a

)

for any

a

∈

D

∖

{

0

D

}

.

•

u

∈

D

is a unit if and only if

ν

⁢

(

u

)

=

ν

⁢

(

1

D

)

.

•

For any

a

∈

D

∖

{

0

D

}

and any unit

u

of

D

, we have

ν

⁢

(

a

)

=

ν

⁢

(

a

⁢

u

)

.

These facts can be proven as follows:

•

If

a

∈

D

∖

{

0

D

}

, then

ν

⁢

(

1

D

)

≤

ν

⁢

(

1

D

⋅

a

)

=

ν

⁢

(

a

)

.

•

If

u

∈

D

is a unit, then let

v

∈

D

be its

inverse

(

http://planetmath.org/MultiplicativeInverse

). Thus,

ν

⁢

(

1

D

)

≤

ν

⁢

(

u

)

≤

ν

⁢

(

u

⁢

v

)

=

ν

⁢

(

1

D

)

.

Conversely, if

ν

⁢

(

u

)

=

ν

⁢

(

1

D

)

, then there exist

q

,

r

∈

D

with

ν

⁢

(

r

)

<

ν

⁢

(

u

)

=

ν

⁢

(

1

D

)

or

r

=

0

D

such that

1

D

=

q

⁢

u

+

r

.

Since

ν

⁢

(

r

)

<

ν

⁢

(

1

D

)

is impossible, we must have

r

=

0

D

. Hence,

q

is the inverse of

u

.

•

Let

v

∈

D

be the inverse of

u

. Then

ν

⁢

(

a

)

≤

ν

⁢

(

a

⁢

u

)

≤

ν

⁢

(

a

⁢

u

⁢

v

)

=

ν

⁢

(

a

)

.

Note that an integral domain is a

Euclidean domain

if and only if it has a Euclidean valuation.

Below are some examples of Euclidean domains and their Euclidean valuations:

•

Any field

F

is a Euclidean domain under the Euclidean valuation

ν

⁢

(

a

)

=

0

for all

a

∈

F

∖

{

0

F

}

.

•

ℤ

is a Euclidean domain with

absolute value

acting as its Euclidean valuation.

•

If

F

is a field, then

F

⁢

[

x

]

, the ring of polynomials over

F

, is a Euclidean domain with degree acting as its Euclidean valuation: If

n

is a nonnegative integer and

a

0

,

…

,

a

n

∈

F

with

a

n

≠

0

F

, then

ν

⁢

(

∑

j

=

0

n

a

j

⁢

x

j

)

=

n

.

Due to the fact that the ring of polynomials over any field is always a Euclidean domain with degree acting as its Euclidean valuation, some refer to a Euclidean valuation as a

degree function

. This is done, for example, in Joseph J. Rotman’s

.

Title

Euclidean valuation

Canonical name

EuclideanValuation

Date of creation

2013-03-22 12:40:45

Last modified on

2013-03-22 12:40:45

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

15

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 13F07

Synonym

degree function

Related topic

PID

Related topic

UFD

Related topic

Ring

Related topic

IntegralDomain

Related topic

EuclideanRing

Related topic

ProofThatAnEuclideanDomainIsAPID

Related topic

DedekindHasseValuation

Related topic

EuclideanNumberField

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