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 dμ ∈

such that

.

μ

|K = .

dμ

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

} .