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 $L/K$ 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) ${{𝒪}}_L/\mathfrak{𝔭}{{𝒪}}_L$ is a field, where $\mathfrak{𝔭}$ is the maximal ideal of ${{𝒪}}_K$.
• (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 $\mathrm{Gal}(\ell /k)$, 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 ramified
• $G$ 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

• Cassels, J.W.S. (1986). Local Fields. London Mathematical Society Student Texts. 3. Cambridge University Press. ISBN 0-521-31525-5. https://books.google.com/books?id=UY52SQnV9w4C&q=%22finite+extension%22. 
• Weiss, Edwin (1976). Algebraic Number Theory (2nd unaltered ed.). Chelsea Publishing. ISBN 0-8284-0293-0. https://books.google.com/books?id=S38pAQAAMAAJ&q=%22finite+extension%22.