Number Theory: An Introduction to Mathematics

286 VI Hensel’sp-adic Numbers

NowE = F(i)containsCand the absolute value onEreduces to the usual
absolute value onC. To prove the theorem it is enough to show thatE =C.For
thenR⊆F⊆CandFhas dimension 1 or 2 as a vector space overRaccording as
i∈/Fori∈F.
Assume on the contrary that there existsζ ∈ E\C. Consider the function
φ:C→Rdefined by

φ(z)=|z−ζ|

and putr=infz∈Cφ(z).Sinceφ( 0 )=|ζ|andφ(z)>|ζ|for|z|> 2 |ζ|, and since

φis continuous, the compact set{z∈C:|z|≤ 2 |ζ|}contains a pointwsuch that

φ(w)=r.

Thus if we putω=ζ−w,thenω=0and

0 <r=|ω|≤|ω−z| for everyz∈C.

We will show that|ω−z|=rfor everyz∈Csuch that|z|<r.

Ifε=e^2 πi/n,then

ωn−zn=(ω−z)(ω−εz)···(ω−εn−^1 z)

and hence

|ωn−zn|≥rn−^1 |ω−z|.

Thus|ω−z|≤r| 1 −zn/ωn|.Since|z|<|ω|, by lettingn→∞we obtain|ω−z|≤r.

But this is possible only if|ω−z|=r.

Thus if 0<|z|<r,thenωmay be replaced byω−z. It follows that|ω−nz|=r

for every positive integern. Hencer ≥n|z|−r, which yields a contradiction for

sufficiently largen. 

If a fieldFis locally compact with respect to anarchimedean absolute value, then

it is certainly complete and so, by Theorem 22, it is equivalent either toRor toCwith

the usual absolute value. It will now be shown that a fieldFis locally compact with

respect to a non-archimedean absolute value if and only if it is a complete field of the

type discussed in Proposition 13. It should be observed that a non-archimedean valued

fieldFis locally compact if and only if its valuation ringRis compact, since then any

closed ball inFis compact.

Proposition 23Let F be a field with a non-archimedean absolute value||.ThenF is

locally compact with respect to the topology induced by the absolute value if and only

if the following three conditions are satisfied:

(i) F is complete,

(ii)the absolute value||is discrete,

(iii)the residue field is finite.

Proof As we have just observed,Fis locally compact if and only if its valuation ring

Ris compact. Moreover, sinceRis a subset of the metric spaceF, it is compact if and

only if any sequence of elements ofRhas a convergent subsequence.