268 VI Hensel’sp-adic Numbers
If we puta=b−^1 c,then|a| 1 >1,|a| 2 <1. This proves the assertion form=2. We
now assumem>2 and use induction. Then there existb,c∈Fsuch that
|b| 1 > 1 , |b|k<1for1<k<m,
|c| 1 > 1 , |c|m< 1.If|b|m<1 we can takea=b.If|b|m=1 we can takea=bncfor sufficiently largen.
If|b|m>1 we can takea=fn(b)cfor sufficiently largen.
Thus for eachi∈{ 1 ,...,m}we can chooseai∈Fso that
|ai|i> 1 , |ai|k<1forallk=i.Then
x=x 1 fn(a 1 )+···+xmfn(am)satisfies the requirements of the proposition for sufficiently largen.
It follows from Proposition 5, that if|| 1 ,...,||mare nontrivial pairwise inequiv-
alent absolute values of a fieldF, then there exists ana∈Fsuch that|a|k> 1 (k=
1 ,...,m). Consequently the absolute values aremultiplicatively independent,i.e.if
ρ 1 ,...,ρmare nonnegative real numbers, not all zero, then for some nonzeroa∈F,
|a|
ρ 1
1 ···|a|ρm
m =^1.3 Completions................................................
Any fieldFwith an absolute value||has the structure of a metric space, with the
metric
d(a,b)=|a−b|,and thus has an associated topology. Since|a|<1 if and only ifan→0asn→∞,
it follows that two absolute values are equivalent if and only if the induced topologies
are the same.
When we use topological concepts in connection with valued fields we will always
refer to the topology induced bythe metric space structure. In this sense addition and
multiplication are continuous operations, since
|(a+b)−(a 0 +b 0 )|≤|a−a 0 |+|b−b 0 |,
|ab−a 0 b 0 |≤|a−a 0 ||b|+|a 0 ||b−b 0 |.Inversion is also continuous at any pointa 0 =0, since if|a−a 0 |<|a 0 |/2then
|a 0 |< 2 |a|and
|a−^1 −a 0 −^1 |=|a−a 0 ||a|−^1 |a 0 |−^1 < 2 |a 0 |−^2 |a−a 0 |.Thus a valued field is atopological field.