View source for Locally compact field

In algebra, a '''locally compact field''' is a [[topological field]] whose topology forms a [[Locally compact space|locally compact]] [[Hausdorff space]].<ref>{{citation|title=Functional Analysis and Valuation Theory|first=Lawrence|last=Narici|publisher=[[CRC Press]]|year=1971|isbn=9780824714840|pages=21–22|url=https://books.google.com/books?id=dSGMFF3viGkC&pg=PA21}}.</ref> These kinds of fields were originally introduced in [[p-adic analysis]] since the fields <math>\mathbb{Q}_p</math> are locally compact topological spaces constructed from the norm <math>|\cdot|_p</math> on <math>\mathbb{Q}</math>. The topology (and metric space structure) is essential because it allows one to construct analogues of [[algebraic number field]]s in the p-adic context.

== Structure ==

=== Finite dimensional vector spaces ===
One of the useful structure theorems for vector spaces over locally compact fields is that the finite dimensional vector spaces have only an equivalence class of norm: the [[sup norm]]<ref name=":0">{{Cite book|last=Koblitz|first=Neil|title=p-adic Numbers, p-adic Analysis, and Zeta-Functions|pages=57–74}}</ref> <sup>pg. 58-59</sup>.

=== Finite field extensions ===
Given a finite field extension <math>K/F</math> over a locally compact field <math>F</math>, there is at most one unique field norm <math>|\cdot|_K</math> on <math>K</math> extending the field norm <math>|\cdot|_F</math>; that is,<blockquote><math>|f|_K = |f|_F </math></blockquote>for all <math>f\in K</math> which is in the image of <math>F \hookrightarrow K</math>. Note this follows from the previous theorem and the following trick: if <math>||\cdot||_1,||\cdot||_2</math> are two equivalent norms, and<blockquote><math>||x||_1 < ||x||_2</math></blockquote>then for a fixed constant <math>c_1</math> there exists an <math>N_0 \in \mathbb{N}</math> such that<blockquote><math>\left(\frac{||x||_1}{||x||_2} \right)^N < \frac{1}{c_1}</math></blockquote>for all <math>N \geq N_0</math> since the sequence generated from the powers of <math>N</math> converge to <math>0</math>.

==== Finite Galois extensions ====
If the index of the extension is of degree <math>n = [K:F]</math> and <math>K/F</math> is a [[Galois extension]], (so all solutions to the minimal polynomial of any <math>a \in K</math> is also contained in <math>K</math>) then the unique field norm <math>|\cdot|_K</math> can be constructed using the [[field norm]]<ref name=":0" /> <sup>pg. 61</sup>. This is defined as<blockquote><math>|a|_K = |N_{K/F}(a)|^{1/n}</math></blockquote>Note the n-th root is required in order to have a well-defined field norm extending the one over <math>F</math> since given any <math>f \in K</math> in the image of <math>F \hookrightarrow K</math> its norm is<blockquote><math>N_{K/F}(f) = \det m_f = f^n</math></blockquote>since it acts as scalar multiplication on the <math>F</math>-vector space <math>K</math>.

== Examples ==

=== Finite fields ===
All finite fields are locally compact since they can be equipped with the discrete topology. In particular, any field with the discrete topology is locally compact since every point is the neighborhood of itself, and also the closure of the neighborhood, hence is compact.

=== Local fields ===
The main examples of locally compact fields are the p-adic rationals <math>\mathbb{Q}_p</math> and finite extensions <math>K/\mathbb{Q}_p</math>. Each of these are examples of [[local field]]s. Note the algebraic closure <math>\overline{\mathbb{Q}}_p</math> and its completion <math>\mathbb{C}_p</math> are '''not''' locally compact fields<ref name=":0" /> <sup>pg. 72</sup> with their standard topology.

==== Field extensions of Q<sub>p</sub> ====
Field extensions <math>K/\mathbb{Q}_p</math> can be found by using [[Hensel's lemma]]. For example, <math>f(x) = x^2 - 7 = x^2 - (2 + 1\cdot 5 )</math> has no solutions in <math>\mathbb{Q}_5</math> since <blockquote><math>\frac{d}{dx}(x^2 - 5) = 2x</math></blockquote>only equals zero mod <math>p</math> if <math>x \equiv 0 \text{ } (p)</math>, but <math>x^2 - 7</math> has no solutions mod <math>5</math>. Hence <math>\mathbb{Q}_5(\sqrt{7})/\mathbb{Q}_5</math> is a quadratic field extension.

==See also==

* {{annotated link|Compact group}}
* {{annotated link|Complete field}}
* {{annotated link|Local field}}
* {{annotated link|Locally compact group}}
* {{annotated link|Locally compact quantum group}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Ramification of local fields}}
* {{annotated link|Topological abelian group}}
* {{annotated link|Topological field}}
* {{annotated link|Topological group}}
* {{annotated link|Topological module}}
* {{annotated link|Topological ring}}
* {{annotated link|Topological semigroup}}
* {{annotated link|Topological vector space}}

==References==
{{reflist}}

== External links ==

* Inequality trick https://math.stackexchange.com/a/2252625

[[Category:Topology]]