Sec. 2] generalization to number fields 17
2. Generalization to number fields
i. The definition.
2.1. Let K be a number field. I.e., K is a finite extension of . It is
well known from algebraic number theory [Cas67] that there is a set Val(K) of
normalized valuations .
ν
on K satisfying the following conditions.
a) The functions .
ν
: K are indeed valuations. I.e., for x, y ,
i) x
ν
0,
ii) x
ν
= 0 if and only if x = 0,
iii) xy
ν
= x
ν
· y ν,
iv) x + y
ν
x
ν
+ y ν.
b) There is the product formula
ν∈Val(K)
x
ν
= 1
for every x K\{0}.
2.2. Further, for L/K a degree d extension of number fields, the sets Val(K)
and Val(L) are compatible in the following sense.
i) For every .
μ
Val(L), there are a valuation .
ν
Val(K) and
such that
.
μ
|K = .

ν
.
In this case, it is said that .
μ
is lying above .
ν
.
ii) For every .
ν
Val(K), there are only a finite number of valuations
.
μ1
, . . . , .
μl
Val(L) lying above . ν. One has
l
i=1
dμi = d .
This implies
x
d
ν
=
l
i=1
x
μi
for every x K.
2.3. A valuation is called Archimedean if it lies above the valuation .

of . Otherwise, it is called non-Archimedean.
If ν is non-Archimedean, then one has the ultrametric triangle inequality
x + y
ν
max{x
ν
, y
ν
} .
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