valuation ring of a field

valuation ring of a field

In this article,

K

is a field with a nontrivial nonarchimedean

absolute value

(

valuation

)

|

⋅

|

and

K

*

its

multiplicative group

of units (nonzero elements).

Proposition 1

.

1.

A

=

d

⁢

f

{

x

∈

K

∣

|

x

|

≤

1

}

is a ring, called the

valuation ring

of

(

K

,

|

⋅

|

)

,

2.

𝔪

=

d

⁢

f

{

x

∈

K

∣

|

x

|

<

1

}

is the unique

maximal ideal

of

A

, and

A

*

=

{

x

∈

K

∣

|

x

|

=

1

}

,

3.

K

is the

fraction field

of

A

.

Proof.

For (1), note that

1

∈

A

, that

x

,

y

∈

A

⇒

|

x

|

≤

1

,

|

y

|

≤

1

⇒

|

x

⁢

y

|

≤

1

⇒

x

⁢

y

∈

A

, and that

x

,

y

∈

A

⇒

|

x

-

y

|

≤

max

⁡

(

|

x

|

,

|

-

y

|

)

≤

1

⇒

x

-

y

∈

A

.

For (2), it is obvious that

𝔪

+

𝔪

⊂

𝔪

and that

𝔪

⁢

A

⊂

𝔪

so that

𝔪

is an ideal. Clearly

A

-

𝔪

=

{

x

∈

K

∣

|

x

|

=

1

}

which is obviously

A

*

and the result follows from general considerations regarding units in a

local ring

.

Finally, to prove (3), choose some

x

∈

K

with

|

x

|

<

1

(to do this, choose any

z

whose valuation is not

1

; then either

z

or

z

-

1

will suffice). Given

y

∈

K

*

, there is some

n

such that

|

y

|

⋅

|

x

|

n

<

1

, so that

y

⁢

x

n

∈

A

and thus

y

⁢

x

n

x

n

=

y

is in the fraction field of

A

.
∎

We say that the absolute value

|

⋅

|

is

discrete

if

|

K

*

|

is a discrete subgroup of

ℝ

>

0

. Note that

ℝ

>

0

≅

(

ℝ

,

+

)

via

log

, so discrete subgroups are

isomorphic

to

ℤ

(are a

lattice

in

ℝ

), and thus a discrete absolute value is of the form

|

K

*

|

=

α

ℤ

for some

α

≥

1

, and

α

=

1

corresponds to the trivial absolute value.

Proposition 2

.

In the notation of the preceding theorem, TFAE:

1.

A

is principal

2.

|

⋅

|

is discrete

3.

A

is

Noetherian

If any of these hold,

A

is a

discrete valuation ring

(DVR).

Proof.

(

1

⇒

2

): If

A

is principal, then

𝔪

=

(

π

)

with

|

π

|

<

1

. Since

A

is a UFD, any element

x

∈

A

-

{

0

}

can be written uniquely as

x

=

u

⁢

π

n

for

u

∈

A

*

,

n

≥

0

, and then

|

x

|

=

|

u

|

⋅

|

π

|

n

=

|

π

|

n

. Thus

|

A

-

{

0

}

|

=

|

π

|

ℕ

and

|

K

*

|

=

|

π

|

ℤ

so that

|

⋅

|

is discrete.

(

2

⇒

1

): If the absolute value is discrete, we may choose

π

∈

K

*

with

|

π

|

<

1

but with the largest possible absolute value strictly less than

1

. Then for

x

∈

𝔪

, we have

|

x

|

<

1

, so

|

x

|

≤

|

π

|

and thus

|

x

π

|

≤

1

so that

x

π

∈

A

. It follows that

x

∈

π

⁢

A

=

(

π

)

, so

A

is principal.

Clearly principal implies Noetherian, so it suffices to prove that

3

⇒

2

: if

|

⋅

|

is not discrete, then

A

is not Noetherian. But if the absolute value is not discrete, we can choose a

convergent sequence

of absolute values and, using the fact that the valuations form an

additive

subgroup

of

ℝ

, we can find a convergent sequence

(

r

n

)

with

r

n

+

1

>

r

n

,

lim

⁡

r

n

=

1

, and a

sequence

of elements of

A

with

|

x

n

|

=

r

n

. Now consider

I

n

=

{

x

∈

A

,

|

x

|

≤

r

n

}

. Then

I

1

⊂

⋯

⊂

I

n

⊂

I

n

+

1

⊂

⋯

and

x

n

+

1

∈

I

n

+

1

\

I

n

, so that

A

is not Noetherian.

The fact that

A

is a DVR follows trivially if any of these conditions holds.
∎

Title

valuation ring of a field

Canonical name

ValuationRingOfAField

Date of creation

2013-03-22 19:03:25

Last modified on

2013-03-22 19:03:25

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11R99

Classification

msc 12J20

Classification

msc 13A18

Classification

msc 13F30

Related topic

HenselianField

Related topic

RingOfExponent

Defines

valuation ring

Defines

discrete valuation

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