Number Theory: An Introduction to Mathematics

266 VI Hensel’sp-adic Numbers

Two absolute values,|| 1 and|| 2 ,onafieldFare said to beequivalentwhen, for
anya∈F,

|a| 1 <1 if and only if|a| 2 < 1.

This implies that|a| 1 >1 if and only if|a| 2 >1 and hence also that|a| 1 =1ifand
only if|a| 2 =1. Thus if one absolute value is trivial, so also is the other. It now follows
from Proposition 3 thattwo absolute values,|| 1 and|| 2 ,on a field F are equivalent if
and only if there exists a real numberρ> 0 such that|a| 2 =|a|ρ 1 for every a∈F.
We have seen that the fieldQof rational numbers admits thep-adic absolute
values||pin addition to the ordinary absolute value||∞. These absolute values are
all inequivalent since, ifpandqare distinct primes,

|p|p< 1 , |p|q= 1 , |p|∞=p> 1.

It was first shown by Ostrowski (1918) that these are essentially the only absolute
values onQ:

Proposition 4Every nontrivial absolute value||of the rational fieldQis equivalent
either to the ordinary absolute value||∞or to a p-adic absolute value||pfor some
prime p.

Proof Letb,cbe integers>1. By writingcto the baseb, we obtain

c=cmbm+cm− 1 bm−^1 +···+c 0 ,

where 0≤cj<b(j= 0 ,...,m)andcm=0. Thenm≤logc/logb,sincecm≥1.
If we putμ=max 1 ≤d<b|d|, it follows from the triangle inequality that

|c|≤μ( 1 +logc/logb){max( 1 ,|b|)}logc/logb.

Takingc=anwe obtain, for anya>1,

|a|≤μ^1 /n( 1 +nloga/logb)^1 /n{max( 1 ,|b|)}loga/logb

and hence, lettingn→∞,

|a|≤{max( 1 ,|b|)}loga/logb.

Suppose first that|a|>1forsomea>1. It follows that|b|>1foreveryb> 1
and

|b|^1 /logb≥|a|^1 /loga.

In fact, sinceaandbmay now be interchanged,

|b|^1 /logb=|a|^1 /loga.

Thusρ=log|a|/logais a positive real number independent ofa>1and|a|=aρ.
It follows that|a|=|a|ρ∞for every rational numbera. Thus the absolute value is
equivalent to the ordinary absolute value.