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\section*{local field}

This article concerns the notion of ``local field'' as it is commonly used in [[algebraic number theory]]. For another notion of ``local field'' in [[commutative algebra]], see [[local field (commutative algebra)]].

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_local_rings_warning}{Relation to local rings (warning)}\dotfill \pageref*{relation_to_local_rings_warning} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{local field} is a [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] (non-[[discrete space|discrete]]) [[topological field]].

Basic examples are the [[p-adic numbers]] $\mathbb{Q}_p$ and the field of [[Laurent series]] $\mathbb{F}_q((t))$ over a [[finite field]] $\mathbb{F}_q$. Local fields are opposite to \emph{[[global fields]]} in that where (under the [[function field analogy]]) the latter may be thought of as fields of [[rational functions]] on [[arithmetic curves]], local fields are like fields of functions on [[formal disks]] inside such curves. Accordingly the [[Langlands correspondence]] for [[global fields]] has a ``localization'' to the [[local Langlands corrrespondence]] for local fields.

Note that for a [[topological field]], the [[closed subspace|topological closure]] of $\{0\}$ is an [[ideal]], which must therefore be either $\{0\}$ or the whole field. It follows that either a topological field is [[separation axioms|T]]$_1$ (and therefore Hausdorff or T$_2$; see [[uniform space]]), or has the [[discrete and codiscrete topology|codiscrete topology]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

A local field $K$ carries a [[valuation]] ${\|-\|}: K \to \mathbb{R}_{\geq 0}$ defined by

\begin{displaymath}
{\|a\|} = \frac{\mu(a X)}{\mu(X)}
\end{displaymath}
where $\mu$ is any [[Haar measure]] defined on the underlying locally compact Hausdorff additive group of $K$, and $X$ is any set such that $0 \lt \mu(X) \lt \infty$.

By analyzing the possibilities for the valuation, any local field is one of the following types:

\begin{itemize}%
\item [[characteristic|Characteristic zero]]. In this case local fields $F$ are [[complete space|completions]] of [[number fields]] with respect to [[metrics]] induced by [[valuations]]. The valuations may be

\begin{itemize}%
\item [[archimedean field|Archimedean]]. Here for every $x \in F$, there exists $n \in \mathbb{N}$ such that ${\|n x\|} \gt 1$, where ${\| \cdot \|}$ is the valuation. The local fields in this case are isomorphic as topological fields to $\mathbb{R}$ or $\mathbb{C}$.


\item [[nonarchimedean field|Nonarchimedean]]. Such valuations are [[discrete valuations]], and are the completions of discrete valuations induced by prime ideals $v$ of the ring of [[algebraic integers]] $\mathcal{O}_k$ in a number field $k$. The valuation on the [[number field]] is defined by ${\|x\|_v} = q^{-n}$ where $q$ is the cardinality of the [[finite field]] $\mathcal{O}_k/v$, and $n$ is the least integer such that $x \in v^n$. The completion is called the \textbf{$v$-adic completion} and is denoted $k_v$.



\end{itemize}

\item [[characteristic|Characteristic]] $p \gt 1$. In this case local fields are fields of [[Laurent series]] $\mathbb{F}_q((t))$ over a finite field $\mathbb{F}_q$ of cardinality $q = p^n$; here ${\|f(t)\|} = q^{-n}$ where $f(t) = a_n t^n + a_{n+1}t^{n+1} + \ldots$. The valuation is nonarchimedean.



\end{itemize}
Local fields are technically useful in modern [[number theory]]; for example in formulating local-to-global principles, and in formulations of [[class field theory]] following Tate's thesis. Part of the technical convenience resides in the fact that one can effectively do [[Fourier analysis]] on them; as additive [[topological abelian group|topological groups]], they are self-dual locally compact abelian groups (in the sense of [[Pontrjagin dual|Pontryagin duality]]).

\hypertarget{relation_to_local_rings_warning}{}\subsection*{{Relation to local rings (warning)}}\label{relation_to_local_rings_warning}

It is possible to construe ``local field'' in at least two other ways, to wit:

\begin{itemize}%
\item As meaning ``[[field of fractions]] of an [[integral domain]] that is a [[local ring]]''.


\item As meaning ``field of fractions of an integral domain that arises as the completion of a local ring with respect to its canonical valuation''.



\end{itemize}
The first meaning is not too serious (and is seldom if ever considered seriously), since usually a field $F$ will not uniquely determine a local subring giving rise to it, nor does this meaning imply any tight connection to local topological conditions such as local compactness. Under this interpretation, $\mathbb{Q}$ would be a ``local field'', which is virtually unheard of.

The second meaning has more content, because the Cauchy completeness (with respect to an $\mathfrak{m}$-topology, where $\mathfrak{m}$ is the maximal ideal of some local ring) determines the local ring via the topology: the complement of $x$ such that $x^{-n}$ converges to $0$. There is nontrivial intersection with the notion of local field as defined above, since the \emph{nonarchimedean} local fields as defined above are conspicuous examples of this second meaning. Observe however that

\begin{itemize}%
\item The archimedean local fields $\mathbb{R}$, $\mathbb{C}$ do \emph{not} arise this way;


\item Under the $m$-[[adic topology]], the completion of a local ring $R$ with maximal ideal $m$, i.e., the [[inverse limit]] of the [[diagram]]

\begin{displaymath}
\ldots R/m^{n+1} \stackrel{proj}{\to} R/m^n \to \ldots \to R/m
\end{displaymath}
is typically not compact (and its field of fractions is not locally compact). It is of course compact if each $R/m^n$ is finite with the discrete topology.



\end{itemize}
In any case, the second meaning certainly occurs in the literature, as in the famous text \emph{Corps Loceaux} by Serre. For more on this, see [[local field (commutative algebra)]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[local Langlands correspondence]]


\item [[higher local field]]


\item [[global field]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Local_field}{Local field}}

\end{itemize}
[[!redirects local field]] [[!redirects local fields]]



\end{document}