2 Equivalence 2652 Equivalence
Ifλ,μ,αare positive real numbers withα<1, then
(
λ
λ+μ)α
+(
μ
λ+μ)α
>λ
λ+μ+
μ
λ+μ= 1
and hence
λα+μα>(λ+μ)α.It follows that if||is an absolute value on a fieldFand if 0<α<1, then||αis also
an absolute value, since
|a+b|α≤(|a|+|b|)α≤|a|α+|b|α.Actually, if||is anon-archimedeanabsolute value on a fieldF, then it follows directly
from the definition that, for anyα>0,||αis also a non-archimedean absolute value
onF.However,if||is anarchimedeanabsolute value onFthen, for all largeα>0,
||αis not an absolute value onF.For| 2 |>1 and hence, ifα>log 2/log| 2 |,
| 1 + 1 |α> 2 =| 1 |α+| 1 |α.Proposition 3Let|| 1 and|| 2 be absolute values on a field F such that|a| 2 < 1 for
any a∈F with|a| 1 < 1 .If|| 1 is nontrivial, then there exists a real numberρ> 0
such that
|a| 2 =|a|ρ 1 for every a∈F.Proof By taking inverses we see that also|a| 2 >1foranya∈ Fwith|a| 1 >1.
Chooseb∈Fwith|b| 1 >1. For any nonzeroa∈Fwe have|a| 1 =|b|γ 1 ,where
γ=log|a| 1 /log|b| 1.Letm,nbe integers withn>0 such thatm/n>γ.Then|a|n 1 <|b|m 1 and hence
|an/bm| 1 <1. Therefore also|an/bm| 2 <1 and by reversing the argument we obtain
m/n>log|a| 2 /log|b| 2.Similarly ifm′,n′are integers withn′>0suchthatm′/n′<γ,then
m′/n′<log|a| 2 /log|b| 2.It follows that
log|a| 2 /log|b| 2 =γ=log|a| 1 /log|b| 1.Thus if we putρ=log|b| 2 /log|b| 1 ,thenρ>0and|a| 2 =|a|ρ 1. This holds trivially
also fora=0.