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 [math]\displaystyle{ L/K }[/math] be a finite Galois extension of nonarchimedean local fields with finite residue fields [math]\displaystyle{ \ell/k }[/math] and Galois group [math]\displaystyle{ G }[/math]. Then the following are equivalent.
- (i) [math]\displaystyle{ L/K }[/math] is unramified.
- (ii) [math]\displaystyle{ \mathcal{O}_L / \mathfrak{p}\mathcal{O}_L }[/math] is a field, where [math]\displaystyle{ \mathfrak{p} }[/math] is the maximal ideal of [math]\displaystyle{ \mathcal{O}_K }[/math].
- (iii) [math]\displaystyle{ [L : K] = [\ell : k] }[/math]
- (iv) The inertia subgroup of [math]\displaystyle{ G }[/math] is trivial.
- (v) If [math]\displaystyle{ \pi }[/math] is a uniformizing element of [math]\displaystyle{ K }[/math], then [math]\displaystyle{ \pi }[/math] is also a uniformizing element of [math]\displaystyle{ L }[/math].
When [math]\displaystyle{ L/K }[/math] is unramified, by (iv) (or (iii)), G can be identified with [math]\displaystyle{ \operatorname{Gal}(\ell/k) }[/math], 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 [math]\displaystyle{ L/K }[/math] be a finite Galois extension of nonarchimedean local fields with finite residue fields [math]\displaystyle{ l/k }[/math] and Galois group [math]\displaystyle{ G }[/math]. The following are equivalent.
- [math]\displaystyle{ L/K }[/math] is totally ramified
- [math]\displaystyle{ G }[/math] coincides with its inertia subgroup.
- [math]\displaystyle{ L = K[\pi] }[/math] where [math]\displaystyle{ \pi }[/math] is a root of an Eisenstein polynomial.
- The norm [math]\displaystyle{ N(L/K) }[/math] contains a uniformizer of [math]\displaystyle{ K }[/math].
See also
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.
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