Krasner’s lemma

Krasner’s lemma

Krasner’s lemma (along with Hensel’s lemma) connects

valuations

on fields to the

algebraic structure

of the fields, and in particular to

polynomial roots

.

Lemma 1

.

(Krasner’s Lemma) Let

K

be a field of

characteristic

0

complete

with respect to a nontrivial nonarchimedean

absolute value

. Assume

α

,

β

∈

K

¯

(where

K

¯

is some

algebraic closure

of

K

) are such that for all nonidentity

embeddings

σ

∈

Hom

K

⁡

(

K

⁢

(

α

)

,

K

¯

)

we have

|

α

-

β

|

<

|

σ

⁢

(

α

)

-

α

|

. Then

K

⁢

(

α

)

⊂

K

⁢

(

β

)

.

This says that for any

α

∈

K

¯

, there is a

neighborhood

of

α

each of whose

elements

generates at least the same field as

α

does.

Proof.

It suffices to show that for every

σ

∈

Hom

K

⁢

(

β

)

⁡

(

K

⁢

(

α

,

β

)

,

K

¯

)

, we have

σ

⁢

(

α

)

=

α

, for then

α

is in the

fixed field

of every embedding of

K

⁢

(

β

)

, so

α

∈

K

⁢

(

β

)

. Note that

|

σ

⁢

(

α

)

-

β

|

=

|

σ

⁢

(

α

)

-

σ

⁢

(

β

)

|

=

|

σ

⁢

(

α

-

β

)

|

=

|

α

-

β

|

where the final equality follows since

|

σ

⁢

(

⋅

)

|

is another absolute value extending

|

⋅

|

K

to

K

⁢

(

α

,

β

)

and thus must be equal to

|

⋅

|

. But then

|

σ

⁢

(

α

)

-

α

|

=

|

(

σ

⁢

(

α

)

-

β

)

+

(

β

-

α

)

|

≤

max

⁡

(

|

σ

⁢

(

α

)

-

β

|

,

|

α

-

β

|

)

=

|

α

-

β

|

But this is impossible by the bounds on

α

,

β

unless

σ

⁢

(

α

)

=

α

.
∎

The first application of Krasner’s lemma is to show that

splitting fields

are “locally constant” in the sense that sufficiently close

polynomials

in

K

⁢

[

X

]

have the same splitting fields.

Proposition 2

.

With

K

as above, let

P

⁢

(

X

)

∈

K

⁢

[

X

]

be a monic

irreducible polynomial

of degree

n

with (distinct) roots

α

1

,

…

,

α

n

. Then any

monic polynomial

Q

⁢

(

X

)

∈

K

⁢

[

X

]

of degree

n

that is “sufficiently close” to

P

⁢

(

X

)

will be

irreducible

over

K

with roots

β

1

,

…

⁢

β

n

, and (after renumbering)

K

⁢

(

α

i

)

=

K

⁢

(

β

i

)

.

Here “sufficiently close” means the following: consider the space of degree

n

polynomials over

K

as

homeomorphic

to

K

n

as a

topological space

; close then means close in the obvious metric

induced

by

|

⋅

|

.

Proof.

Since

P

⁢

(

X

)

has distinct roots, we may choose

0

<

γ

<

min

⁡

(

|

α

i

-

α

j

|

)

for

i

≠

j

≤

n

. Since the roots of a polynomial vary continuously with its

coefficients

, we say that a degree

n

polynomial

Q

⁢

(

X

)

∈

K

⁢

[

X

]

is sufficiently close to

P

⁢

(

X

)

if

Q

⁢

(

X

)

has roots

β

1

,

…

,

β

n

with

|

α

i

-

β

i

|

<

γ

. But

{

α

j

}

j

≠

i

are all the

Galois conjugates

of

α

i

, and

|

α

i

-

β

i

|

<

γ

<

|

α

i

-

α

j

|

by construction, so by Krasner’s lemma,

K

⁢

(

α

i

)

⊂

K

⁢

(

β

i

)

. But

[

K

(

β

i

)

:

K

]

≤

deg

Q

=

deg

P

=

[

K

(

α

i

)

:

K

]

so that

K

⁢

(

β

i

)

=

K

⁢

(

α

i

)

. In

addition

, we see that

deg

Q

=

[

K

(

β

i

)

:

K

]

and thus that

Q

⁢

(

X

)

is irreducible.
∎

We use this fact to show that every

finite extension

of

ℚ

p

arises as a

completion

of some

number field

.

Corollary 3

.

Let

K

be a finite extension of

Q

p

of degree

n

. Then there is a number field

E

and an absolute value

|

⋅

|

on

E

such that

E

^

≅

K

.

Proof.

Let

K

=

ℚ

p

⁢

(

α

)

and let

P

be the

minimal polynomial

for

α

over

ℚ

p

. Since

ℚ

is dense in

ℚ

p

, we can choose

Q

⁢

(

X

)

∈

ℚ

⁢

[

X

]

(note: in

ℚ

⁢

[

X

]

, not

ℚ

p

⁢

[

X

]

), and

β

a root of

Q

⁢

(

X

)

, as in the

proposition

, so that

ℚ

p

⁢

(

α

)

=

ℚ

p

⁢

(

β

)

. Let

E

=

ℚ

⁢

(

β

)

. Clearly

E

is a number field which, when regarded as embedded in

ℚ

p

⁢

(

β

)

, has absolute value

|

⋅

|

E

, the

restriction

of the absolute value on

ℚ

p

⁢

(

α

)

=

ℚ

p

⁢

(

β

)

. Then

E

^

is a complete field with respect to that absolute value;

ℚ

p

⁢

(

β

)

is as well, and

E

is dense in both, so we must have

E

^

=

ℚ

p

⁢

(

β

)

=

ℚ

p

⁢

(

α

)

=

K

.
∎

Finally, we can prove the following

generalization

of Krasner’s Lemma, which is also given that name in the literature:

Lemma 4

.

Let

K

be a field of characteristic

0

complete with respect to a nontrivial nonarchimedean absolute value, and

K

¯

an algebraic closure of

K

. Extend the absolute value on

K

to

K

¯

; this

extension

is unique. Let

K

¯

^

be the completion of

K

¯

with respect to this absolute value. Then

K

¯

^

is

algebraically closed

.

Proof.

Let

α

be algebraic over

K

¯

^

and

P

⁢

(

X

)

its monic irreducible polynomial in

K

¯

^

⁢

[

X

]

. Since

K

¯

is dense in

K

¯

^

, by proposition

2

we may choose

Q

⁢

(

x

)

∈

K

¯

⁢

[

X

]

with a root

β

∈

K

¯

^

such that

K

¯

^

⁢

(

α

)

=

K

¯

^

⁢

(

β

)

. But

K

¯

^

⁢

(

β

)

=

K

¯

^

so that

α

∈

K

¯

^

.
∎

Title

Krasner’s lemma

Canonical name

KrasnersLemma

Date of creation

2013-03-22 19:03:02

Last modified on

2013-03-22 19:03:02

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 12J99

Classification

msc 11S99

Classification

msc 13H99

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