Cryptography Reference
In-Depth Information
n
Definition 5.1.15
The
affine coordinate ring
over
k
of an affine algebraic set
X
⊆ A
defined over
k
is
k
[
X
]
= k
[
x
1
,...,x
n
]
/I
k
(
X
)
.
k
Warning:
Here
[
X
] does not denote polynomials in the variable
X
. Hartshorne and Fulton
write
A
(
X
) and
(
X
) respectively for the affine coordinate ring.
Exercise 5.1.16
Prove that
k
[
X
] is a commutative ring with an identity.
Note that
k
[
X
] is isomorphic to the ring of all functions
f
:
X
→ k
given by polynomials
defined over
.
Hilbert's Nullstellensatz is a powerful tool for understanding
I
k
(
X
) and it has several
other applications (e.g., we use it in Se
ct
ion
7.5
). We follow the presentation of Fulton [
199
].
Note that it is necessary to work over
k
k
.
n
be an affi
n
e algebraic set defined over
T
heorem 5.1.17
(Weak Nullstellensatz) Let X
⊆ A
k
and let
m
be a maximal ide
a
l of the affine coordinate ring
k
[
X
]
. Then V
(m)
={
P
}
for
some P
=
(
P
1
,...,P
n
)
∈
X
(
k
)
and
m
=
(
x
1
−
P
1
,...,x
n
−
P
n
)
.
Proof
See Section 1.7 of Fulton [
199
].
k
= ∅
Corollary 5.1.18
If I is a proper ideal in
[
x
1
,...,x
n
]
then V
(
I
)
.
We can now state the
Hilbert Nullstellensatz
. This form of the theorem (which applies
to
I
k
(
V
(
I
)) where
k
is not necessarily algebraically closed, appears as Proposition VIII.7.4
of [
271
].
Theorem 5.1.19
Let I be an ideal in R
= k
[
x
1
,...,x
n
]
. Then I
k
(
V
(
I
))
=
rad
R
(
I
)
(see
Section
A.9
for the definition of the radical ideal).
Proof
See Section 1.7 of Fulton [
199
].
Co
ro
llary 5.1.20
Let f
(
x,y
)
∈ k
[
x,y
]
be irred
uc
ible over
k
and let X
=
V
(
f
(
x,y
))
⊂
A
2
(
k
)
. Then I
(
X
)
=
(
f
(
x,y
))
, i.e. the ideal over
k
[
x,y
]
generated by f
(
x,y
)
.
Proof
By Theorem
5.1.19
we have
I
(
X
)
[
x,y
] is a unique factori-
sation domain and
f
(
x,y
) is irreducible, then
f
(
x,y
)isp
ri
me.So
g
(
x,y
)
=
rad
k
((
f
(
x,y
))). Since
k
∈
rad
k
((
f
(
x,y
)))
implies
g
(
x,y
)
n
=
∈ k
|
f
(
x,y
)
h
(
x,y
)forsome
h
(
x,y
)
[
x,y
], which implies
f
(
x,y
)
∈
g
(
x,y
) and
g
(
x,y
)
(
f
(
x,y
)).
5.2 Projective algebraic sets
Studying affine algebraic sets is not sufficient for our applications. In particular, the set of
affine points of the Weierstrass equation of an elliptic curve (see Section
7.2
) does not form
a group. Projective geometry is a way to “complete” the picture by adding certain “points
at infinity”.
