Local Fields

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 27 November 2014

Local Fields

A discrete valuation on a field F is a surjective map v:F(), such that:

(i) v(0)=,
(ii) v:F× is a surjective homomorphism,
(iii) v(x+y)inf{v(x),v(y)} for all x,yF.

Examples

(1) (t-adic valuation) Consider rational functions in t over a field F. Every non-zero rational function f(t)F(t) can be written in the form f(t)=tna(t)b(t) where a(t) and b(t) are polynomials with non-zero constant term and n. Putting v(f(t))=n gives a discrete valuation on F(t).
(2) (p-adic valuation) Fix a prime p. Every non-zero rational number f can be written in the form f=pnab where a and b are integers relatively prime to p and n. Putting v(f)=n gives a discrete valuation on .

Let R be an integral domain. A function v:R(), satisfying (i)-(iii) above is called a discrete valuation on R. A discrete valuation on an integral domain extends uniquely to a discrete valuation on its quotient field.

Examples

(1) The t-adic valuation v on F[t], where v(a(t)) is the highest power of t dividing a(t)F[t] extends to the t-adic valuation on F(t).
(2) The p-adic valuation v on , where v(a) is the highest power of p dividing a extends to the p-adic valuation on .

Given a field F and a discrete valuation v on F we set 𝔇 = {xF:v(x)0}, 𝔅 = {xF:v(x)1}, U = {xF:v(x)=0}. Then 𝔇 is a ring, called the valuation ring. It is a local ring with maximal ideal 𝔅. The set U is the group of units of 𝔇. The field k=𝔇/𝔅 is called the residue class field associated to the valuation. An element π in F with valuation v(π)=1 is called a uniformizer. The ring 𝔇 is principal ideal domain and the non-zero ideals of 𝔇 are 𝔅n=(πn)={xF:v(x)n}.

Example

(1) In the rational function case the valuation ring 𝔇 is the localization of F[t] at the ideal (t) generated by t. The residue class field is isomorphic to F. The canonical map is given by evaluation at t=0.
(2) In the p-adic case 𝔇 is the localization of at the ideal (p). The residue class field is the finite field with p elements.

Exercise. An element x of F can be expressed uniquely in the form x=uπn, uU a unit and n.

Fix ρ>1. We define a function :F[0,), from F to the non-negative real numbers by x=1ρv(x) forxF. This satisfies

(1) x=0 if and only if x=0.
(2) xy=xy for all x,yF.
(3) x+ymax{x,y} for all x,yF.

The last inequality implies

(3)' x+yx+y.

A function satisfying (1), (2) and (3) is called a non-archimedean absolute value. Function which satisfy (1), (2) and (3)' but not (3) are called archimedean absolute values. An absolute value gives a metric d(x,y)=x-y on F. If it is non-archimedean we have d(x,z)max{d(x,y),d(y,z)}.

The non-archimedean metric defined by v makes F into a topological field. Note the topology does not depend on the choice of ρ. In this topology the open balls about aF are the sets {xF:v(x-a)>n} and the closed balls are the sets {xF:v(x-a)n}. But {xF:v(x-a)>n}={xF:v(x-a)n+1}. Thus each open ball is closed. Hence each open set is closed, i.e the topology is totally disconnected.

If F is complete with respect to the metric we call v a complete discrete valuation. Since different choices of ρ>1 give equivalent metrics this notion is well defined. We can always complete of F with respect to the metric . The Cauchy sequences in F form a ring under pointwise operations. The null sequences form a maximal ideal. The completion F of F is the quotient field. The field F is naturally embedded in the quotient field via the constant sequences. The the valuation F extends uniquely to the completion. In fact if a is determined by Cauchy sequence {an}, the sequence {v(an)} will be eventually constant and v(α)=limnv(an) is well defined.

The valuation ring of the completion is the closure of the valuation ring 𝔇 of F in F. The residue class rings of F and F completion are naturally isomorphic.

Let π be a uniformiser of F and A a set of residue class representatives in 𝔇 of k. Then all sums n-anπn, with each anA, converge in F. Every element of F is uniquely represented by such a sum.

The valuation ring of the completion is all the sums n=0anπn.

Examples

(1) The completion of F[t] with respect to the t-adic valuation is the ring of formal power series in t with coefficients in F, F[[t]]= { n=0 antn:an F } . The completion of F(t) is F((t)), the field of formal Laurent series with a pole at zero, F((t))= { n- antn:an F } . The valuation is given by v ( n- antn=min {n:an0}. )
(2) The completion of with respect to the p-adic topology is the ring of p-adic integers, p= { n=0 anpn:an {0,1,,p-1} } . The completion of , is the field of p-adic numbers, p= { n- anpn:an {0,1,,p-1} } . The valuation is given by v(n-anpn) =min{n:an0}.

Local Field. A field which is complete with respect to a complete valuation and has finite residue class field is called is called a local field.

(1) A local field of characteristic p>0 is of the form F((t)), with F finite of characteristic p.
(2) The local fields of characteristic zero are the finite algebraic extensions of p.

We can make a canonical choice of absolute value in a local field F. Let q denote the order of its residue class field. Then define x=1qv(x) forxF.

The valuation ring of a local field 𝔇, and its ideals 𝔅n, are compact. They form a compact system of neighbourhoods of zero in F. Hence F is a locally compact Hausdorff topological group under addition. We can fix a Haar measure μ on F by setting μ(𝔇)=1.

For all xF, and measurable A, we have μ(xA)=xμ(A).

Proof. (Sketch)

This trivial for x=0. For x0, μ(A)=μ(xA) is also a Haar measure. Thus μ(xA)=c(x)μ(A) for some positive real c(x) Now deduce c(xy)=c(x)c(y) for all x,yF. Then by taking A=𝔇 show that c(u)=1 for uU, and (count cosets), c(π)=1q. The result now follows.

The sets 1+𝔅n, n>0 form a compact system of neighbourhoods of the identity in F×. Thus F× is a locally compact Hausdorff topological group under multiplication.

Haar measure on F× is given by dxx.

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