Number Theory: An Introduction to Mathematics

4 Non-Archimedean Valued Fields 273

4 Non-ArchimedeanValuedFields...............................

Throughout this section we denote by F a field with a non-archimedean absolute value
||.A basic property of such fields is the following simple lemma. It may be interpreted
as saying that in ultrametric geometry every triangle is isosceles.

Lemma 8If a,b∈F and|a|<|b|,then|a+b|=|b|.

Proof We certainly have

|a+b|≤max{|a|,|b|} = |b|.

On the other hand, sinceb=(a+b)−a,wehave

|b|≤max{|a+b|,|−a|}

and, since|−a|=|a|<|b|, this implies|b|≤|a+b|.

It may be noted that ifa=0andb=−a,then|a|=|b|and|a+b|<|b|.From
Lemma 8 it follows by induction that ifa 1 ,...,an∈Fand|ak|<|a 1 |for 1<k≤n,
then

|a 1 +···+an|=|a 1 |.

As an application we show thatif a field E is a finite extension of a field F,then
the trivial absolute value on E is the only extension to E of the trivial absolute value
on F. By Proposition 2, any extension toEof the trivial absolute value onFmust be
non-archimedean. Supposeα∈Eand|α|>1. Thenαsatisfies a polynomial equation

αn+cn− 1 αn−^1 +···+c 0 = 0

with coefficientsck∈F.Since|ck|=0 or 1 and since|αk|<|αn|ifk<n, we obtain
the contradiction|αn|=|αn+cn− 1 αn−^1 +···+c 0 |=0.
As another application we prove

Proposition 9If a field F has a non-archimedean absolute value||, then the
valuation on F can be extended to the polynomial ring F[t]by defining the absolute
value of f(t)=a 0 +a 1 t+···+antnto be|f|=max{|a 0 |,...,|an|}.

Proof We need only show that|fg|=|f||g|, since it is evident that|f|=0ifand
only iff=0andthat|f+g|≤|f|+|g|.Letg(t)=b 0 +b 1 t+···+bmtm.Then
f(t)g(t)=c 0 +c 1 t+···+cltl,where

ci=a 0 bi+a 1 bi− 1 +···+aib 0.

Ifr is the least integer such that|ar|=|f|ands the least integer such that
|bs|=|g|,thenarbshas strictly greatest absolute value among all productsajbk
withj+k=r+s. Hence|cr+s|=|ar||bs|and|fg|≥|f||g|. On the other hand,

|fg|=max

i

|ci|≤max

j,k

|aj||bk|=|f||g|.

Consequently|fg|=|f||g|. Clearly also|f|=|a|if f =a∈F. (The absolute
value onFcan be further extended to the fieldF(t)of rational functions by defining
|f(t)/g(t)|to be|f|/|g|.)