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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)?.

Contents

Definition

A local field is a locally compact Hausdorff (non-discrete) topological field.

Note that for a topological field, the topological closure of {0}\{0\} is an ideal, which must therefore be either {0}\{0\} or the whole field. It follows that either a topological field is T1_1 (and therefore Hausdorff or T2_2; see uniform space), or has the codiscrete topology.

Properties

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

a=μ(aX)μ(X){\|a\|} = \frac{\mu(a X)}{\mu(X)}

where μ\mu is any Haar measure defined on the underlying locally compact Hausdorff additive group of KK, and XX is any set such that 0<μ(X)<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 FF are completions of number fields with respect to metrics induced by valuations. The valuations may be

    • Archimedean. Here for every xFx \in F, there exists nn \in \mathbb{N} such that nx>1{\|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 vv of the ring of algebraic integers 𝒪 k\mathcal{O}_k in a number field kk. The valuation on the number field is defined by x v=q n{\|x\|_v} = q^{-n} where qq is the cardinality of the finite field 𝒪 k/v\mathcal{O}_k/v, and nn is the least integer such that xv nx \in v^n. The completion is called the vv-adic completion and is denoted k vk_v.

  • Characteristic p>1p \gt 1. In this case local fields are fields of Laurent series 𝔽 q((t))\mathbb{F}_q((t)) over a finite field 𝔽 q\mathbb{F}_q of cardinality q=p nq = p^n; here f(t)=q n{\|f(t)\|} = q^{-n} where f(t)=a nt n+a n+1t n+1+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).

Relation to local rings (warning)

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 FF 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 𝔪\mathfrak{m}-topology, where 𝔪\mathfrak{m} is the maximal ideal of some local ring) determines the local ring via the topology: the complement of xx such that x nx^{-n} converges to 00. 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 mm-adic topology, the completion of a local ring RR with maximal ideal mm, i.e., the inverse limit of the diagram

    R/m n+1projR/m nR/m\ldots R/m^{n+1} \stackrel{proj}{\to} R/m^n \to \ldots \to R/m

    is typically not compact (and its field of fractions is not locally compact). It is of course compact if each R/m nR/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)?.

Revised on February 4, 2013 10:18:50 by Urs Schreiber (82.113.99.102)