Cryptography Reference
In-Depth Information
In particular, if x,y
∈ k
(
C
)
are separating elements then ∂x/∂y
=
1
/
(
∂y/∂x
)
=
0
.
Let t
∈ k
(
C
)
. Then ∂t/∂x
=
0
if and only if t is not a separating element.
Proof
See Lemma IV.1.6 of Stichtenoth [
529
].
1
Exercise 8.5.15
Let
C
= P
over
F
p
with variable
x
and let
δ
(
f
)
=
∂f/∂x
. Show that
δ
(
x
p
)
=
0.
(
C
) we can introduce the differentials on
a curve over a field. Our definition is purely formal and the symbol
dx
is not assumed to
have any intrinsic meaning. We essentially follow Section IV.1 of Stichtenoth [
529
]; for a
slightly different approach see Section 8.4 of Fulton [
199
].
Now we have defined
∂f/∂x
for general
f
∈ k
Definition 8.5.16
Let
C
be a curve over
k
.Thesetof
differentials
k
(
C
) (some authors
write
1
k
(
C
)) is the quotient of the free
k
(
C
)-module on symbols
dx
for
x
∈ k
(
C
) under
the relations:
1.
dx
0if
x
is a separating element.
2. If
x
is a separating element and
h
1
,h
2
∈ k
=
(
C
) then
h
1
dx
+
h
2
dx
=
(
h
1
+
h
2
)
dx
.
3. If
x
is a separating element and
y
∈ k
(
C
) then
dy
=
(
∂y/∂x
)
dx
.
In other words, differentials are equivalence classes of formal symbols
m
(
C
)
h
i
dx
i
:
x
i
,h
i
∈ k
i
=
1
where one may assume the
x
i
are all separating elements.
k
∈ k
Lemma 8.5.17
Let C be a curve over
and x,y
(
C
)
separating elements.
1. dx
=
0
if x is not a separating element.
2. d
(
x
+
y
)
=
dx
+
dy.
3. d
(
λx
)
=
λdx and dλ
=
0
for all λ
∈ k
.
4. d
(
xy
)
ydx.
5. If x is a separating element and y
=
xdy
+
∈ k
(
C
)
then dx
+
dy
=
(1
+
(
∂y/∂x
))
dx.
, d
(
x
n
)
nx
n
−
1
dx.
6. For n
∈ Z
=
xdy
)
/y
2
.
7. d
(
x/y
)
=
(
ydx
−
8. If f
∈ k
(
C
)
then d
(
f
(
x
))
=
(
∂f/∂x
)
dx.
f
(
x
)
iy
i
−
1
dy.
10. If F
(
x,y
)
is a rational function in x and y then dF
(
x,y
)
, d
(
f
(
x
)
y
i
)
(
∂f/∂x
)
y
i
dx
9. For i
∈ Z
=
+
=
+
(
∂F/∂x
)
dx
(
∂F/
∂y
)
dy.
Exercise 8.5.18
Prove Lemma
8.5.17
.
