Finite extensions of local fields explained

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 be a finite Galois extension of nonarchimedean local fields with finite residue fields \ell/k and Galois group G . Then the following are equivalent.

• (i)

L/K is unramified.

• (ii)

l{O} _{L} / ak{p} l{O} _{L} is a field, where ak{p} is the maximal ideal of l{O} _{K} .

• (iii)

[L : K] = [\ell : k]

• (iv) The inertia subgroup of

G is trivial.

• (v) If

\pi is a uniformizing element of K , then \pi is also a uniformizing element of L .

When

L/K is unramified, by (iv) (or (iii)), G can be identified with \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 be a finite Galois extension of nonarchimedean local fields with finite residue fields l/k and Galois group G . The following are equivalent.L/K is totally ramifiedG coincides with its inertia subgroup.L = K[\pi] where \pi is a root of an Eisenstein polynomial.

• The norm

N(L/K) contains a uniformizer of K .

See also

• Abhyankar's lemma
• Unramified morphism

References

• Book: Cassels, J.W.S. . J. W. S. Cassels

. J. W. S. Cassels . Local Fields . London Mathematical Society Student Texts . 3 . . 1986 . 0-521-31525-5 . 0595.12006 .

• Book: Weiss, Edwin . Algebraic Number Theory . . 2nd unaltered . 1976 . 0-8284-0293-0 . 0348.12101 .