Cryptography Reference
In-Depth Information
4. V
(
fg
)
=
V
(
f
)
∪
V
(
g
)
.
5. V
(
f
)
∩
V
(
g
)
=
V
(
f,g
)
.
Exercise 5.1.8
Prove Proposition
5.1.7
.
The following result assumes a knowledge of Galois theory. See Section
A.7
for back-
ground.
Lemma 5.1.9
Let X
=
V
(
S
)
be an algebraic set with
S
⊆ k
[
x
]
(i.e., X is a
k
-algebraic
k
be an algebraic extension of
k
)
.ForP
set). Let
k
. Let σ
∈
Gal(
k
/
=
(
P
1
,...,P
n
)
define
σ
(
P
)
=
(
σ
(
P
1
)
,...,σ
(
P
n
))
.
1. If P
∈
X
(
k
)
then
σ
(
P
)
∈
X
(
k
)
.
k
)
k
)
2. X
(
={
P
∈
X
(
k
):
σ
(
P
)
=
P for all σ
∈
Gal(
k
/
}
.
Exercise 5.1.10
Prove Lemma
5.1.9
.
k
⊆ A
n
(
k
Definition 5.1.11
The
ideal
over
of a set
X
)is
I
k
(
X
)
={
f
∈ k
[
x
]:
f
(
P
)
=
0 for all
P
∈
X
(
k
)
}
.
(
X
).
1
An algebraic set
X
is
defined over
We define
I
(
X
)
=
I
k
k
(sometimes abbreviated to “
X
over
k
”) if
I
(
X
) can
be generated by elements of
k
[
x
].
Perhaps surprisingly, it is not necessarily true that an algebraic set described by poly-
nomials defined over
. In Remark
5.3.7
we will explain
that these concepts are equivalent for the objects of interest in this topic.
k
is an algebraic set defined over
k
Exercise 5.1.12
Show that
I
k
(
X
)
=
I
(
X
)
∩ k
[
x
].
⊆ A
n
be sets and J a
k
Proposition 5.1.13
Let X,Y
[
x
]
-ideal. Then:
1. I
k
(
X
)
is a
k
[
x
]
-ideal.
2. X
⊆
V
(
I
k
(
X
))
.
3. If X
⊆
Y then I
k
(
Y
)
⊆
I
k
(
X
)
.
4. I
k
(
X
I
k
(
Y
)
.
5. If X is an algebraic set defined over
∪
Y
)
=
I
k
(
X
)
∩
k
then V
(
I
k
(
X
))
=
X.
6. If X and Y are algebraic sets defined over
k
and I
k
(
X
)
=
I
k
(
Y
)
then X
=
Y.
7. J
⊆
I
k
(
V
(
J
))
.
8. I
k
(
∅
)
= k
[
x
]
.
Exercise 5.1.14
Prove Proposition
5.1.13
.
1
The notation
I
k
(
X
) is not standard (Silverman [
505
] calls it
I
(
X/
k
)), but the notation
I
(
X
) agrees with many elementary topics
on algebraic geometry, since they work over an algebraically closed field.