Finite extensions of local fields

In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups.

In this article, a local field is non-archimedean and has finite residue field.

Unramified extension

Let

L

/

K

{\displaystyle L/K}

be a finite Galois extension of nonarchimedean local fields with finite residue fields

ℓ

/

k

{\displaystyle \ell /k}

and Galois group

G

{\displaystyle G}

. Then the following are equivalent.

(i)

L

/

K

{\displaystyle L/K}

is unramified.

(ii)

O

L

/

p

O

L

{\displaystyle {\mathcal {O}}_{L}/{\mathfrak {p}}{\mathcal {O}}_{L}}

is a field, where

p

{\displaystyle {\mathfrak {p}}}

is the maximal ideal of

O

K

{\displaystyle {\mathcal {O}}_{K}}

.

(iii)

[

L

:

K

]

=

[

ℓ

:

k

]

{\displaystyle [L:K]=[\ell :k]}

(iv) The inertia subgroup of

G

{\displaystyle G}

is trivial.

(v) If

π

{\displaystyle \pi }

is a uniformizing element of

K

{\displaystyle K}

, then

π

{\displaystyle \pi }

is also a uniformizing element of

L

{\displaystyle L}

.

When

L

/

K

{\displaystyle L/K}

is unramified, by (iv) (or (iii)), G can be identified with

Gal

⁡

(

ℓ

/

k

)

{\displaystyle \operatorname {Gal} (\ell /k)}

, which is finite cyclic.

The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.

Totally ramified extension

Again, let

L

/

K

{\displaystyle L/K}

be a finite Galois extension of nonarchimedean local fields with finite residue fields

l

/

k

{\displaystyle l/k}

and Galois group

G

{\displaystyle G}

.