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 not. 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. 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:

There is an equivalent definition of non-archimedean local field: it is a field that is complete with respect to a discrete valuation and whose residue field is finite. However, some authors consider a more general notion, requiring only that the residue field be perfect, not necessarily finite.[2] This article uses the former definition.

Induced absolute value[edit]

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 |·| : KR by[3]

|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). This definition is very similar to that of the modular function.

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.[4] 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.

Non-archimedean local field theory[edit]

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

Every non-zero element a of F can be written as a = ϖnu with u a unit, and n a unique integer. The normalized valuation of F is the surjective function v : FZ ∪ {∞} defined by sending a non-zero a to the unique integer n such that a = ϖnu with u a unit, and by sending 0 to ∞. If q is the cardinality of the residue field, the absolute value on F induced by its structure as a local field is given by[5]

|a|=q^{-v(a)}.

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.

Examples[edit]

  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 am 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.

Higher unit groups[edit]

The nth higher unit group of a non-archimedean local field F is

U^{(n)}=1+\mathfrak{m}^n=\left\{u\in\mathcal{O}^\times:u\equiv1\, (\mathrm{mod}\,\mathfrak{m}^n)\right\}

for n ≥ 1. The group U(1) is called the group of principal units, and any element of it is called a principal unit. The full unit group \mathcal{O}^\times is denoted U(0).

The higher unit groups provide a decreasing filtration of the unit group

\mathcal{O}^\times\supseteq U^{(1)}\supseteq U^{(2)}\supseteq\cdots

whose quotients are given by

\mathcal{O}^\times/U^{(n)}\cong\left(\mathcal{O}/\mathfrak{m}^n\right)^\times\text{ and }\,U^{(n)}/U^{(n+1)}\approx\mathcal{O}/\mathfrak{m}

for n ≥ 1.[6] (Here "\approx" means a non-canonical isomorphism.)

Structure of the unit group[edit]

The multiplicative group of non-zero elements of a non-archimedean local field F is isomorphic to

F^\times\cong(\varpi)\times\mu_{q-1}\times U^{(1)}

where q is the order of the residue field, and μq−1 is the group of (q−1)st roots of unity (in F). Its structure as an abelian group depends on its characteristic:

  • If F has positive characteristic p, then
F^\times\cong\mathbf{Z}\oplus\mathbf{Z}/{(q-1)}\oplus\mathbf{Z}_p^\mathbf{N}
where N denotes the natural numbers;
  • If F has characteristic zero (i.e. it is a finite extension of Qp of degree d), then
F^\times\cong\mathbf{Z}\oplus\mathbf{Z}/(q-1)\oplus\mathbf{Z}/p^a\oplus\mathbf{Z}_p^d
where a ≥ 0 is defined so that the group of p-power roots of unity in F is \mu_{p^a}.[7]

Higher-dimensional local fields[edit]

Main article: Higher local field

It is natural to introduce non-archimedean local fields in a uniform geometric way as the field of fractions of the completion of the local ring of a one-dimensional arithmetic scheme of rank 1 at its non-singular point. For generalizations, a local field is sometimes called a one-dimensional local field.

For a non-negative integer n, an n-dimensional local field is a complete discrete valuation field whose residue field is an (n − 1)-dimensional local field.[8] Depending on the definition of local field, a zero-dimensional local field is then either a finite field (with the definition used in this article), or a quasi-finite field,[9] or a perfect field.

From the geometric point of view, n-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an n-dimensional arithmetic scheme.

See also[edit]

Notes[edit]

  1. ^ Page 20 of Weil 1995
  2. ^ See, for example, definition 1.4.6 of Fesenko & Vostokov 2002
  3. ^ Page 4 of Weil 1995
  4. ^ Corollary 1, page 5 of Weil 1995
  5. ^ Weil 1995, chapter I, theorem 6
  6. ^ Neukirch 1999, p. 122
  7. ^ Neukirch 1999, theorem II.5.7
  8. ^ Definition 1.4.6 of Fesenko & Vostokov 2002
  9. ^ Serre 1995

References[edit]

Further reading[edit]

External links[edit]