# Locally compact field

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In algebra, a '''locally compact field''' is a [topological field](/source/topological_field) whose topology forms a [locally compact](/source/Locally_compact_space) [Hausdorff space](/source/Hausdorff_space).<ref>{{citation|title=Functional Analysis and Valuation Theory|first=Lawrence|last=Narici|publisher=[CRC Press](/source/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](/source/p-adic_analysis) since the fields <math>\mathbb{Q}_p</math> of [p-adic number](/source/p-adic_number)s are locally compact topological spaces constructed from the norm <math>|\cdot|_p</math> on <math>\mathbb{Q}</math>. The topology (and [metric space](/source/metric_space) structure) is essential because it allows one to construct analogues of [algebraic number field](/source/algebraic_number_field)s in the p-adic context.

== Structure ==

=== Finite dimensional vector spaces ===
One of the useful structure theorems for [vector space](/source/vector_space)s over locally compact fields is that the finite dimensional vector spaces have only one [equivalence class](/source/equivalence_class) of norms: the [sup norm](/source/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](/source/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 extension is of degree <math>n = [K:F]</math> and <math>K/F</math> is a [Galois extension](/source/Galois_extension), (so all solutions to the [minimal polynomial](/source/Minimal_polynomial_(field_theory)), or [conjugate elements](/source/Conjugate_element_(field_theory)), of any <math>a \in K</math> are also contained in <math>K</math>) then the unique field norm <math>|\cdot|_K</math> can be constructed using the [field norm](/source/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](/source/scalar_multiplication) on the <math>F</math>-vector space <math>K</math>.

== Examples ==

=== Finite fields ===
All [finite field](/source/finite_field)s are locally compact since they can be equipped with the [discrete topology](/source/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](/source/local_field)s. Note the [algebraic closure](/source/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](/source/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|Complete field}}
* {{annotated link|Locally compact group}}
* {{annotated link|Ramification of local fields}}
* {{annotated link|Topological abelian group}}
* {{annotated link|Topological group}}
* {{annotated link|Topological ring}}

==References==
{{reflist}}

== External links ==

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

Category:Topological algebra
Category:Topological groups
Category:Compactness (mathematics)
Category:Field theory

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Adapted from the Wikipedia article [Locally compact field](https://en.wikipedia.org/wiki/Locally_compact_field) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Locally_compact_field?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.