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
Note that for a topological field, the topological closure of is an ideal, which must therefore be either or the whole field. It follows that either a topological field is T (and therefore Hausdorff or T; see uniform space), or has the codiscrete topology.
A local field carries a valuation defined by
where is any Haar measure defined on the underlying locally compact Hausdorff additive group of , and is any set such that .
By analyzing the possibilities for the valuation, any local field is one of the following types:
Archimedean. Here for every , there exists such that , where is the valuation. The local fields in this case are isomorphic as topological fields to or .
Nonarchimedean. Such valuations are discrete valuations, and are the completions of discrete valuations induced by prime ideals of the ring of algebraic integers in a number field . The valuation on the number field is defined by where is the cardinality of the finite field , and is the least integer such that . The completion is called the -adic completion and is denoted .
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 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 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, would be a “local field”, which is virtually unheard of.
The second meaning has more content, because the Cauchy completeness (with respect to an -topology, where is the maximal ideal of some local ring) determines the local ring via the topology: the complement of such that converges to . 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 , do not arise this way;
is typically not compact (and its field of fractions is not locally compact). It is of course compact if each 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)?.