18 the concept of a height [Chap. I
2.4. Definition. Let K be a number field of degree d. Then, for
(x0 : . . . : xn) Pn(K), one puts
Hnaive(x0 : . . . : xn) :=
ν∈Val(K)
max
i=0,...,n
xi
ν
1
d
.
This height function is the number field version of the naive height on Pn. It is
usually called the absolute height.
2.5. Lemma. Let K be a number field, and let (x0 : . . . : xn)
Pn(K).
Further, let L K be a finite extension.
Then the absolute height Hnaive(x0 : . . . : xn) remains unchanged when
(x0 : . . . : xn) is considered as an L-rational point.
Proof. Put d := [L : K]. Then, by the properties of the valuations, we have
Hnaive(x0 : . . . : xn) =
ν∈Val(K)
max
i=0,...,n
xi
ν
1
d
=
ν∈Val(K)
μ∈Val(L)
μ above ν
max
i=0,...,n
xi
μ
1
dd
=
μ∈Val(L)
max
i=0,...,n
xi
μ
1
dd
,
which is exactly the formula for Hnaive(x0 : . . . : xn) considered as an
L-rational point.
2.6. Proposition (D. G. Northcott). Let B, D . Then
{x
Pn(
) | x
Pn(K)
for [K : ] D and Hnaive(x) B}
is a finite set.
Proof. We may work with the number fields K of a fixed degree d.
For x, we choose homogeneous coordinates such that some coordinate equals 1.
Then it is clear that, for every valuation .
ν
and every index i, we have
max{x0 ν, . . . , xn ν} max{1, xi ν} .
Multiplying over all ν and taking the d-th root therefore shows
Hnaive(x0 : . . . : xn) Hnaive(1 : xi) .
Hence, it suffices to verify that the set
{(1 : x)
P1(
) | (1 : x)
P1(K)
for [K : ] = d and Hnaive(1 : x) B}
is finite.
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