288 VI Hensel’sp-adic Numbers
‖a‖ 0 =max
1 ≤i≤n|αi|is a norm onE. Since the fieldFis locally compact, it is also complete. Hence, by
Lemma 7, there exist positive real constantsσ,μsuch that
σ‖a‖ 0 ≤‖a‖≤μ‖a‖ 0 for everya∈E.Consequently, if{ak}is a bounded sequence of elements ofEthen, for each j ∈
{ 1 ,...,n}, the corresponding coefficients{αkj}form a bounded sequence of elements
ofF. Hence, sinceFis locally compact, there exists a subsequence{akv}such that
each of the sequences{αkvj}converges inF, with limitβjsay(j= 1 ,...,n).Itfol-
lows that the subsequence{akv}converges inEwith limitb=β 1 e 1 +···+βnen. Thus
Eis locally compact.
Suppose next thatEis infinite-dimensional overF. Since the valuation onFis
nontrivial, there existsα∈Fsuch thatr=|α|satisfies 0<r<1. LetVbe any
finite-dimensional subspace ofE,letu′∈E\Vand let
d=inf
v∈V‖u′−v‖.SinceVis locally compact,d>0andd=‖u′−v′‖for somev′∈V. Choosek∈Z
so thatrk+^1 <d≤rkand putw′=α−k(u′−v′).Foranyv∈V,
‖αkv+v′−u′‖≥dand hence
‖w′−v‖≥dr−k>r.On the other hand,
‖w′‖=dr−k≤ 1.We now define a sequence{wm}of elements ofEin the following way. Taking
V ={O}we obtain a vectorw 1 withr <‖w 1 ‖≤1. Suppose we have defined
w 1 ,...,wm∈Eso that, for 1≤j≤m,‖wj‖≤1and‖wj−vj‖>rfor allvj
in the vector subspaceVj− 1 ofEspanned byw 1 ,...,wj− 1. Then, takingV =Vm,
we obtain a vectorwm+ 1 such that‖wm+ 1 ‖≤1and‖wm+ 1 −vm+ 1 ‖>rfor all
vm+ 1 ∈Vm. Thus the process can be continued indefinitely. Since‖wm‖≤1forallm
and‖wm−wj‖>rfor 1≤j<m, the bounded sequence{wm}has no convergent
subsequence. ThusEis not locally compact.
Proposition 27A non-archimedean valued field E with zero characteristic is locally
compact if and only if, for some prime p, E is isomorphic to a finite extension of the
fieldQpof p-adic numbers.
Proof IfEis a finite extension of thep-adic fieldQpthen, sinceQpis locally com-
pact, so also isE, by Lemma 26.
Suppose on the other hand thatEis a locally compact valued field with zero char-
acteristic. ThenQ⊆E. By Proposition 23, the residue fieldk=R/Mis finite and