Cryptography Reference
In-Depth Information
We claim the minimal polynomial of
x
over
k
(
θ
)isgivenby
F
(
T
)
=
a
(
T
)
−
θb
(
T
)
.
To see this, first note that
F
(
x
)
=
0. Now,
a
(
T
)
−
θb
(
T
) is irreducible in
k
[
θ,T
] since it
k
is linear in
θ
. The irreducibility of
F
(
T
)in
(
θ
)[
T
] then follows from the Gauss Lemma
(see, for example, Lemma III.6.13 of Hungerford [
271
]).
Exercise 8.1.10
Let
C
1
:
y
2
x
3
and
C
2
:
Y
2
=
=
X
over a field
k
of characteristic not
equal to 2 and consider the map
φ
:
C
1
→
C
2
such that
φ
(
x,y
)
=
(
x,y/x
). Show that
deg(
φ
)
=
1.
Exercise 8.1.11
Let
C
1
:
y
2
x
6
2
x
2
1 and
C
2
:
Y
2
X
3
=
+
+
=
+
2
X
+
1 over a field
k
of
(
x
2
,y
).
characteristic not equal to 2 and consider the map
φ
:
C
1
→
C
2
such that
φ
(
x,y
)
=
Show that deg(
φ
)
=
2.
Exercise 8.1.12
Let
C
1
,C
2
and
C
3
be curves over
k
and let
ψ
:
C
1
→
C
2
and
φ
:
C
2
→
C
3
be morphisms over
k
. Show that deg(
φ
◦
ψ
)
=
deg(
φ
)deg(
ψ
).
Lemma 8.1.13
Let C
1
and C
2
be curves over
k
(in particular, smooth and projective). Let
φ
:
C
1
→
C
2
be a birational map over
k
. Then φ has degree 1.
For Lemma
8.1.15
(and Lemma
8.2.3
) we need the following technical result. This is
a special case of weak approximation; see Stichtenoth [
529
] for a presentation that uses
similar techniques to obtain most of the results in this chapter.
and let Q,Q
∈
Lemma 8.1.14
L
et
C be a curve over
k
C
(
k
)
be distinct points. Then there
is a function f
∈ k
(
C
)
such that v
Q
(
f
)
=
0
and v
Q
(
f
)
>
0
.
Proof
By Lemma
7.1.17
we have
O
Q
,
k
⊆
O
Q,
k
(and vice versa). Hence, there exists
a function
u
∈
O
Q,
k
−
O
Q
,
k
. Then
v
Q
(
u
)
≥
0 while
v
Q
(
u
)
<
0. If
u
(
Q
)
=−
1 then set
u
2
)elseset
f
f
=
1
/
(1
+
=
1
/
(1
+
u
). Then
v
Q
(
f
)
=
0 and
v
Q
(
f
)
>
0 as required.
Lemma 8.1.15
Let C
1
and C
2
be curves over
k
(in particular, smooth and projective). Let
φ
:
C
1
→
C
2
be a rational map over
k
of degree 1. Then φ is an isomorphism.
Proof
Corollary II.2.4 of Silverman [
505
].
8.2 Extensions of valuations
Let
φ
:
C
1
→
C
2
be a non-constant morphism of curves over
k
. Then
F
1
= k
(
C
1
)
isafinite
φ
∗
(
extension of
F
2
=
) under
φ
and a notion of multiplicity of preimages of
Q
(namely, ramification indices). The main
result is Theorem
8.2.9
.
There are several approaches to these results in the literature. One method, which unifies
algebraic number theory and the theory of curves, is to note that if
U
is an open subset
of
C
then
k
(
C
2
)). We now study the preimages of points
Q
∈
C
2
(
k
k
[
U
] is a Dedekind domain. The splitting of the maximal ideal m
Q
of
k
[
U
]
