270 VI Hensel’sp-adic Numbers
Let(an)be a fundamental sequence which is not inN. Then there existsμ> 0
such that|av|≥μfor infinitely manyv.Since|am−an|<μ/2forallm,n≥N,it
follows that|an|>μ/2foralln≥N.Putbn=a−n^1 ifan=0,bn=0ifan=0.
Then(bn)is a fundamental sequence since, form,n≥N,
|bm−bn|=|(an−am)/aman|≤ 4 μ−^2 |an−am|.Since( 1 )−(bnan)∈N, the ideal generated by(an)andN contains the constant
sequence (1) and hence every sequence inF. Since this holds for each sequence
(an)∈F\N, the idealN is maximal.
Consequently (see Chapter I,§8) the quotientF ̄ =F/N is a field. Since (0) is
the only constant sequence inN, by mapping each constant sequence into the coset
ofN which contains it we obtain a field inF ̄isomorphic toF. Thus we may regard
Fas embedded inF ̄.
It follows from Lemma 1(iii), and from the completeness of the field of real
numbers, that|A|=limn→∞|an|exists for any fundamental sequenceA=(an).
Moreover,
|A|≥ 0 , |AB|=|A||B|, |A+B|≤|A|+|B|.Furthermore|A|=0 if and only ifA∈N. It follows that|B|=|C|ifB−C∈N,
since
|B|≤|B−C|+|C|=|C|≤|C−B|+|B|=|B|.Thus we may consider||as defined onF ̄=F/N, and it is then an absolute value
on the fieldF ̄which coincides with the original absolute value when restricted to the
fieldF.
IfA=(an)is a fundamental sequence, and ifAmis the constant sequence(am),
then|A−Am|can be made arbitrarily small by takingmsufficiently large. It follows
thatFisdenseinF ̄,i.e.foranyα∈F ̄and anyε>0 there existsa∈Fsuch that
|α−a|<ε.
We show finally that F ̄ iscompleteas a metric space, i.e. every fundamental
sequence of elements ofF ̄converges to an element ofF ̄.Forlet(αn)be a funda-
mental sequence inF ̄.SinceFis dense inF ̄, for eachnwe can choosean∈Fso that
|αn−an|< 1 /n.Since
|am−an|≤|am−αm|+|αm−αn|+|αn−an|,it follows that(an)is also a fundamental sequence. Thus there existsα∈F ̄such that
limn→∞|an−α|=0. Since
|αn−α|≤|αn−an|+|an−α|,we have also limn→∞|αn−α|=0. Thus the sequence(αn)converges toα.
Summing up, we have proved
Proposition 6If F is a field with an absolute value||, then there exists a fieldF ̄
containing F , with an absolute value||extending that of F , such thatF is complete ̄
and F is dense inF. ̄