Tricomi–Carlitz polynomials: Difference between revisions
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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]] | |||
{{numtheory-stub}} | |||
Latest revision as of 19:17, 17 September 2013
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.
Let be a finite Galois extension of nonarchimedean local fields with finite residue fields and Galois group G.
Unramified extension
The following are equivalent.
- (i) is unramified.
- (ii) is a field, where is the maximal ideal of .
- (iii)
- (iv) The inertia subgroup of G is trivial.
- (v) If is a uniformizing element of , then is also a uniformizing element of .
When is unramified, by (iv) (or (iii)), G can be identified with , 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
The following are equivalent.
- is totally ramified
- coincides with its inertia subgroup.
- where is a root of an Eisenstein polynomial.
- The norm contains a uniformizer of .
See also
References
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