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

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

Let <math>L/K</math> be a finite Galois extension of nonarchimedean local fields with finite residue fields <math>l/k</math> and Galois group ''G''.
<!--== Introduction ==

To motivate the below, we start with how one reduces a global case to a local case. Thus, let ''A'' be a Dedekind domain with the field of fractions ''K'' and <math>L/K</math> a finite separable extension. Let <math>\mathfrak q \in \operatorname{Spec}B</math> {{Citation needed|date=February 2012}} and <math>\mathfrak p = A \cap \mathfrak q</math>. A basic result in [[algebraic number theory]] is that the extension <math>L/K</math> is unramified at <math>\mathfrak{q}</math> if and only if <math>\mathfrak{q}</math> does not divide the [[different ideal|different]] <math>\mathfrak{D}_{B/A}</math> of <math>B</math> over <math>A</math>. (Upon taking a norm, this says that <math>L/K</math> is unramified at <math>\mathfrak{q} \cap A</math> if and only if <math>\mathfrak{q} \cap A</math> does not divide the discriminant of <math>B</math> over <math>A</math>.) Since the different commutes with localization and completion, this reduces to the case when <math>A, B</math> are complete; i.e., <math>L, K</math> are local fields.-->
== Unramified extension ==
The following are equivalent.
*(i) <math>L/K</math> is '''unramified'''.
*(ii) <math>\mathcal{O}_L / \mathcal{O}_L \mathfrak{p}</math> is a field, where <math>\mathfrak{p}</math> is the maximal ideal of <math>\mathcal{O}_K</math>.
*(iii) <math>[L : K] = [l : k]</math>
*(iv) The [[inertia subgroup]] of ''G'' is trivial.
*(v) If <math>\pi</math> is a [[uniformizing element]] of <math>K</math>, then <math>\pi</math> is also a uniformizing element of <math>L</math>.

When <math>L/K</math> is unramified, by (iv) (or (iii)), ''G'' can be identified with <math>\operatorname{Gal}(l/k)</math>, which is finite [[cyclic group|cyclic]].

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

== Totally ramified extension ==

The following are equivalent.
* <math>L/K</math> is '''totally ramified'''
* <math>G</math> coincides with its inertia subgroup.
* <math>L = K[\pi]</math> where <math>\pi</math> is a root of an [[Eisenstein polynomial]].
* The norm <math>N(L/K)</math> contains a uniformizer of <math>K</math>.

== See also ==
*[[Abhyankar's lemma]]

==References==
{{reflist}}
* {{cite book | first=J.W.S. | last=Cassels | authorlink=J. W. S. Cassels | title=Local Fields | series=London Mathematical Society Student Texts | volume=3 | publisher=[[Cambridge University Press]] | year=1986 | isbn=0-521-31525-5 | zbl=0595.12006 }}
* {{cite book | last=Weiss | first=Edwin | title=Algebraic Number Theory | publisher=[[Chelsea Publishing]] | edition=2nd unaltered | year=1976 | isbn=0-8284-0293-0 | zbl=0348.12101  }}

[[Category:Algebraic number theory]]

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