Local field

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In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.[1] Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is Archimedean and those in which it is non-Archimedean. In the first case, one calls the local field an archimedean local field, in the second case, one calls it a non-archimedean local field. There is an equivalent definition of non-archimedean local field given below. Local fields arise naturally in number theory as completions of global fields.

Every local field is isomorphic (as a topological field) to one of the following:

Contents

[edit] Non-Archimedean local field theory

For a non-archimedean local field F (with absolute value denoted by |·|), the following objects are very important:

  • its ring of integers \mathcal{O}\ which is its closed unit ball \{a\in F: |a|\leq 1\} (it is compact),
  • the units in its ring of integers \mathcal{O}^\times which is its unit sphere \{a\in F: |a|= 1\},
  • the unique prime ideal in its ring of integers \mathfrak{m} which is its open unit ball \{a\in F: |a|< 1\},
  • its residue field k=\mathcal{O}/\mathfrak{m} which is finite (since it is compact and discrete).

One often talks about the (discrete) valuation of a non-archimedean local field. This is a map v:F\rightarrow\mathbb{R}\cup\{\infty\} obtained as follows: there is a real number 0 < c < 1 such that

c^{v(a)}=|a|\mbox{ for all }a\in F.

One generally chooses c such that v surjects onto \mathbb{Z}\cup\{\infty\}, and calls this the normalized valuation.

An equivalent definition of a non-archimedean local field is that it is a field that is complete with respect to a discrete valuation and whose residue field is finite.

[edit] Examples

  1. The p-adic numbers: the ring of integers of Qp is the ring of p-adic integers Zp. Its prime ideal is pZp and its residue field is Z/pZ. Every non-zero element of Qp can be written as u pn where u is a unit in Zp and n is an integer, then v(u pn) = n for the normalized valuation.
  2. The formal Laurent series over a finite field: the ring of integers of Fq((T)) is the ring of formal power series Fq[[T]]. Its prime ideal is (T) (i.e. the power series whose constant term is zero) and its residue field is Fq. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:
    v\left(\sum_{i=-m}^\infty a_iT^i\right) = -m (where a−m is non-zero).
  3. The formal Laurent series over the complex numbers is not a local field. For example, its residue field is C[[T]]/(T) = C, which is not finite.

[edit] Induced absolute value

Given a locally compact topological field K, an absolute value can be defined as follows. First, consider the additive group of the field. As a locally compact topological group, it has a unique (up to positive scalar multiple) Haar measure μ. The absolute value is defined so as to measure the change in size of a set after multiplying it by an element of K. Specifically, define |·| : K → R by[2]

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

for any measurable subset X of K (with 0 < μ(X) < ∞). This absolute value does not depend on X nor on the choice of Haar measure (since the same scalar multiple ambiguity will occur in both the numerator and the denominator).

Given such an absolute value on K, a new induced topology can be defined on K. This topology is the same as the original topology.[3] Explicitly, for a positive real number m, define the subset Bm of K by

B_m:=\{ a\in K:|a|\leq m\}.

Then, the Bm make up a neighbourhood basis of 0 in K.

[edit] See also

[edit] Notes

  1. ^ Page 20 of Weil 1995
  2. ^ Page 4 of Weil 1995
  3. ^ Corollary 1, page 5 of Weil 1995

[edit] References

[edit] Further reading

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