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)?.
symmetric monoidal (∞,1)-category of spectra
A local field is a locally compact Hausdorff (non-discrete) topological field.
Note that for a topological field, the 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 T$_1$ (and therefore Hausdorff or T$_2$; see uniform space), or has the codiscrete topology.
A local field $K$ carries a valuation ${\|-\|}: K \to \mathbb{R}_{\geq 0}$ defined by
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:
Characteristic zero. In this case local fields $F$ are completions of number fields with respect to metrics induced by valuations. The valuations may be
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 $\mathbb{C}$.
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 $v$-adic completion and is denoted $k_v$.
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.
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 groups, they are self-dual locally compact abelian groups (in the sense of Pontryagin duality).
It is possible to construe “local field” in at least two other ways, to wit:
As meaning “field of fractions of an integral domain that is a local ring”.
As meaning “field of fractions of an integral domain that arises as the completion of a local ring with respect to its canonical valuation”.
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 nonarchimedean local fields as defined above are conspicuous examples of this second meaning. Observe however that
The archimedean local fields $\mathbb{R}$, $\mathbb{C}$ do not arise this way;
Under the $m$-adic topology, the completion of a local ring $R$ with maximal ideal $m$, i.e., the inverse limit of the diagram
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.
In any case, the second meaning certainly occurs in the literature, as in the famous text Corps Loceaux by Serre. For more on this, see local field (commutative algebra)?.