Number Theory: An Introduction to Mathematics

6 Locally Compact Valued Fields 287

The fieldFis certainly complete if it is locally compact, since any fundamental
sequence is bounded. If the residue field is infinite, then there exists an infinite
sequence(ak)of elements ofRsuch that|ak−aj|=1forj=k. Since the sequence
(ak)has no convergent subsequence,Ris not compact. If the absolute value||is not
discrete, then there exists an infinite sequence(ak)of elements ofRwith

|a 1 |<|a 2 |<···

and|ak|→1ask→∞.Ifk>j,then|ak−aj|=|ak|and again the sequence(ak)
has no convergent subsequence. Thus the conditions (i)–(iii) are all necessary forFto
be locally compact.
Suppose now that the conditions (i)–(iii) are all satisfied and letσ =(ak)be a
sequence of elements ofR. In the notation of Proposition 13, let

ak=

∑

n≥ 0

α(nk)πn,

whereα(nk)∈S.SinceSis finite, there existsα 0 ∈Ssuch thatα( 0 k)=α 0 for infinitely

manyak∈σ.Ifσ 0 is the subsequence ofσcontaining thoseakfor whichα( 0 k)=α 0 ,

then there existsα 1 ∈Ssuch thatα 1 (k)=α 1 for infinitely manyak∈σ 0. Similarly, if

σ 1 is the subsequence ofσ 0 containing thoseakfor whichα( 1 k)=α 1 , then there exists

α 2 ∈Ssuch thatα 2 (k)=α 2 for infinitely manyak∈σ 1. And so on. Ifa(j)∈σj,then

a(j)=α 0 +α 1 π+···+αjπj+

∑

n≥ 0

αn(j)πj+^1 +n.

Buta=

∑

n≥ 0 αnπ

n∈F,sinceFis complete, and|a(j)−a|≤|π|j+ (^1). Thus the
subsequence(a(j))ofσconverges toa.

Corollary 24The fieldQpof p-adic numbers is locally compact, and the ringZpof
p-adic integers is compact.
Corollary 25If K is a finite field, then the field K((t))of all formal Laurent series is
locally compact, and the ring K[[t]]of all formal power series is compact.
We now show that all locally compact valued fieldsFwith a non-archimedean
absolute value can in fact be explicitly determined. It is convenient to treat the cases
whereFhas prime characteristic and zero characteristic separately, since the argu-
ments in the two cases are quite different.
Lemma 26Let F be a locally compact valued field with a nontrivial valuation. A
normed vector space E over F is locally compact if and only if it is finite-dimensional.
Proof Suppose first thatEis finite-dimensional overF.Ife 1 ,...,enis a basis for the
vector spaceE,thenanya∈Ecan be uniquely represented in the form
a=α 1 e 1 +···+αnen,
whereα 1 ,...,αn∈F,and