Number Theory: An Introduction to Mathematics

272 VI Hensel’sp-adic Numbers

Proof Lete 1 ,...,enbe a basis for the vector spaceE.Thenanya ∈ Ecan be
uniquely represented in the form

a=α 1 e 1 +···+αnen,

whereα 1 ,...,αn∈F. It is easily seen that

‖a‖ 0 =max

1 ≤i≤n

|αi|

is a norm onE, and it is sufficient to prove the proposition for‖‖ 2 =‖‖ 0 .Since

‖a‖ 1 ≤‖a‖ 0 (‖e 1 ‖ 1 +···+‖en‖ 1 ),

we can takeσ=(‖e 1 ‖ 1 +···+‖en‖ 1 )−^1. To establish the existence ofμwe assume
n>1 and use induction, since the result is obviously true forn=1.
Assume, contrary to the assertion, that there exists a sequencea(k)∈Esuch that

‖a(k)‖ 1 <εk‖a(k)‖ 0 ,

whereεk>0andεk→0ask→∞. We may suppose, without loss of generality, that

|α(nk)|=‖a(k)‖ 0

and also, by replacinga(k)by(αn(k))−^1 a(k),thatαn(k)=1. Thusa(k)=b(k)+en,where

b(k)=α( 1 k)e 1 +···+α(nk−) 1 en− 1 ,

and‖a(k)‖ 1 →0ask→∞. The sequencesαi(k)(i= 1 ,...,n− 1 )are fundamental
sequences inF,since

‖b(j)−b(k)‖ 1 ≤‖b(j)+en‖ 1 +‖b(k)+en‖ 1 =‖a(j)‖ 1 +‖a(k)‖ 1

and, by the induction hypothesis,

|αi(j)−αi(k)|≤μn− 1 ‖b(j)−b(k)‖ 1 (i= 1 ,...,n− 1 ).

Hence, sinceFis complete, there existαi∈Fsuch that|α(ik)−αi|→ 0 (i= 1 ,...,
n− 1 ).Put

b=α 1 e 1 +···+αn− 1 en− 1.

Since‖b(k)−b‖ 1 ≤σn−−^11 ‖b(k)−b‖ 0 , it follows that‖b(k)−b‖ 1 →0. But ifa=b+en,
then

‖a‖ 1 ≤‖a−a(k)‖ 1 +‖a(k)‖ 1 =‖b−b(k)‖ 1 +‖a(k)‖ 1.

Lettingk→∞, we obtaina=0, which contradicts the definition ofa.