Number Theory: An Introduction to Mathematics

2 Equivalence 267

Suppose next that|a|≤1foreverya>1 and so for everya∈Z. Since the
absolute value onQis nontrivial, we must have|a|<1 for some integera=0. The
setMof alla∈Zsuch that|a|<1 is a proper ideal inZand hence is generated by
an integerp>1. We will show thatpmust be a prime. Supposep=bc,whereb
andcare positive integers. Since|b||c|=|p|<1, we may assume without loss of
generality that|b|<1. Thenb∈Mand thusb=pdfor somed∈Z. Hencecd= 1
and soc=1. Thusphas no nontrivial factorization.
Every rational numbera=0 can be expressed in the forma=pvb/c,wherevis
an integer andb,care integers not divisible byp. Hence|b|=|c|=1and|a|=|p|v.
We can write|p|=p−ρ, for some real numberρ>0. Then|a|=p−vρ=|a|
ρ
p,and
thus the absolute value is equivalent to thep-adic absolute value.

Similarly, the absolute values on the fieldF=K(t)considered in example (iii)
of§1 are all inequivalent and it may be shown that any nontrivial absolute value onF
whose restriction toKis trivial is equivalent to one of these absolute values.
In example (ii) of§1 we have made a specific choice in each class of equivalent
absolute values. The choice which has been made ensures the validity of theproduct
formula: for any nonzeroa∈Q,

|a|∞

∏

p

|a|p= 1 ,

where|a|p=1 for at most finitely manyp.
Similarly, in example (iii) of§1 the absolute values have been chosen so that, for
any nonzero f ∈ K(t),|f|∞

∏

p|f|p =1, where|f|p =1 for at most finitely
manyp.
The followingapproximation theorem, due to Artin and Whaples (1945), treats
several absolute values simultaneously. Forp-adic absolute values of the rational field
Qthe result also follows from the Chinese remainder theorem (Corollary II.38).

Proposition 5Let|| 1 ,...,||mbe nontrivial pairwise inequivalent absolute values of
an arbitrary field F and let x 1 ,...,xmbe any elements of F. Then for each realε> 0
there exists an x∈F such that

|x−xk|k<ε for 1 ≤k≤m.

Proof During the proof we will more than once use the fact that if fn(x) =
xn( 1 +xn)−^1 ,then|fn(a)|→0or1asn→∞according as|a|<1or|a|>1.
We show first thatthere exists an a∈F such that

|a| 1 > 1 , |a|k< 1 for 2 ≤k≤m.

Since|| 1 and|| 2 are nontrivial and inequivalent, there existb,c∈Fsuch that

|b| 1 < 1 , |b| 2 ≥ 1 ,

|c| 1 ≥ 1 , |c| 2 < 1.