proof of Ostrowski’s valuation theorem

proof of Ostrowski’s valuation theorem

This article proves Ostrowski’s theorem on

valuations

of

ℚ

, which states:

Theorem 1

.

(Ostrowski) Over

Q

, every nontrivial

absolute value

is

equivalent

either to

|

⋅

|

p

for some prime

p

, or to the usual absolute value

|

⋅

|

∞

.

We start with an estimation lemma:

Lemma 2

.

If

m

,

n

>

1

are

integers

and

|

⋅

|

any nontrivial absolute value on

Q

, then

|

m

|

≤

max

(

1

,

|

n

|

)

log

⁡

m

/

log

⁡

n

.

Proof.

Write

m

=

a

0

+

a

1

⁢

n

+

⋯

+

a

r

⁢

n

r

for

a

i

∈

ℤ

,

0

≤

a

i

≤

n

-

1

, and with

a

r

≠

0

. Then clearly

|

a

i

|

=

|

1

+

⋯

+

1

⏟

a

i

|

≤

a

i

⁢

|

1

|

=

a

i

≤

n

by the

triangle inequality

; also,

r

<

log

⁡

m

log

⁡

n

.

Thus

|

m

|

=

|

a

0

+

a

1

n

+

⋯

+

a

r

n

r

|

≤

(

r

+

1

)

n

max

(

1

,

|

n

|

)

r

≤

(

1

+

log

⁡

m

log

⁡

n

)

n

max

(

1

,

|

n

|

)

log

⁡

m

/

log

⁡

n

Replace

m

by

m

t

for

t

a

positive integer

, and take

t

th

roots of the resulting

inequality

, to get

|

m

|

≤

(

1

+

t

log

⁡

m

log

⁡

n

)

1

/

t

n

1

/

t

max

(

1

,

|

n

|

)

log

⁡

m

/

log

⁡

n

Now let

t

→

∞

; the first two factors each approach

1

, and the lemma follows.

Proof of Ostrowski’s theorem:

First assume that for every

n

>

1

we have

|

n

|

>

1

. Then by the lemma,

|

m

|

≤

|

n

|

log

⁡

m

/

log

⁡

n

, so that for every

m

,

n

we have

|

m

|

1

/

log

⁡

m

≤

|

n

|

1

/

log

⁡

n

Since this holds for every

m

,

n

>

0

, after reversing the roles of

m

,

n

, we see that in fact equality holds, so that for every

m

,

|

m

|

1

/

log

⁡

m

=

c

and

|

m

|

=

c

log

⁡

m

for some constant

c

; this absolute value is obviously equivalent to

|

m

|

∞

=

e

log

⁡

m

.

If instead, for some

n

>

1

we have

|

n

|

<

1

, then by the lemma, for every

m

,

|

m

|

≤

1

. Thus the absolute value is nonarchimedean. Define

A

=

{

x

∈

ℚ

∣

|

x

|

≤

1

}

and let

𝔪

⊂

A

be the (unique)

maximal ideal

defined by

𝔪

=

{

x

∈

ℚ

∣

|

x

|

<

1

}

. Then

ℤ

⊂

A

since

|

m

|

≤

1

for every

m

, and

𝔪

∩

ℤ

is nonzero since otherwise the valuation would be trivial (we would have

|

m

|

=

1

for every

m

). Thus

𝔪

∩

ℤ

is prime since

𝔪

is, so is equal to

(

p

)

for some

rational prime

p

. Now, if

p

∤

a

for an integer

a

, then

|

a

|

cannot be strictly less than

1

(else it would be in

(

p

)

), so

|

a

|

=

1

and

a

∈

A

⋆

. But given any

x

∈

ℚ

, we can write

x

=

a

⁢

p

t

b

with

a

,

b

prime to

p

, so that

|

x

|

=

|

a

|

⋅

|

p

|

t

|

b

|

=

|

p

|

t

so that the valuation is obviously equivalent to the

p

-adic valuation.

Title

proof of Ostrowski’s valuation theorem

Canonical name

ProofOfOstrowskisValuationTheorem

Date of creation

2013-03-22 17:58:26

Last modified on

2013-03-22 17:58:26

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Proof

Classification

msc 13A18

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