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.
<!--== 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 <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 == Let <math>L/K</math> be a finite Galois extension of nonarchimedean local fields with finite residue fields <math>\ell/k</math> and Galois group <math>G</math>. Then the following are equivalent. *(i) <math>L/K</math> is '''unramified'''. *(ii) <math>\mathcal{O}_L / \mathfrak{p}\mathcal{O}_L </math> is a field, where <math>\mathfrak{p}</math> is the maximal ideal of <math>\mathcal{O}_K</math>. *(iii) <math>[L : K] = [\ell : k]</math> *(iv) The inertia subgroup of <math>G</math> 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}(\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>L/K</math> be a finite Galois extension of nonarchimedean local fields with finite residue fields <math>l/k</math> and Galois group <math>G</math>. 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 *Unramified morphism
==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 |url=https://books.google.com/books?id=UY52SQnV9w4C&q=%22finite+extension%22}} * {{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 |url=https://books.google.com/books?id=S38pAQAAMAAJ&q=%22finite+extension%22}}
Category:Algebraic number theory