Cryptography Reference
In-Depth Information
5
Varieties
The purpose of this chapter is to state some basic definitions and results from algebraic
geometry that are required for the main part of the topic. In particular, we define algebraic
sets, irreducibility, function fields, rational maps and dimension. The chapter is not intended
as a self-contained introduction to algebraic geometry. Many proofs are omitted. We make
the following recommendations to the reader:
1. Readers who want a very elementary introduction to elliptic curves are advised to consult
one or more of Koblitz [
313
], Silverman and Tate [
508
], Washington [
560
], Smart [
513
]
or Stinson [
532
].
2. Readers who wish to learn algebraic geometry properly should first read a basic
text such as Reid [
447
] or Fulton [
199
]. They can then skim this chapter and con-
sult Stichtenoth [
529
], Moreno [
395
], Hartshorne [
252
], Lorenzini [
355
] or Shafare-
vich [
489
] for detailed proofs and discussion.
5.1 Affine algebraic sets
Let
k
be a perfect field contained in a fixed a
lg
ebraic closure
k
. All algebraic extensions
k
/
k
are implicitly assumed to be subfields of
k
. We use the notation
k
[
x
]
= k
[
x
1
,...,x
n
]
(in later sections we also use
k
[
x
]
= k
[
x
0
,...,x
n
]). When
n
=
2 or 3 we often write
k
[
x,y
]
or
k
[
x,y,z
].
Define
affine
n
-space over
n
(
n
. We call
1
(
2
(
k
as
A
k
)
= k
A
k
)the
affine line
and
A
k
)
k ⊆ k
then we have the natural inclusion
n
(
n
(
k
). We
the
affine plane
o
ver
k
.If
A
k
)
⊆ A
n
for
n
(
n
(
n
.
write
A
A
k
) and so
A
k
)
⊆ A
Definition 5.1.1
Let
S
⊆ k
[
x
]. Define
n
(
V
(
S
)
={
P
∈ A
k
):
f
(
P
)
=
0 for all
f
∈
S
}
.
If
S
={
f
1
,...,f
m
}
then we write
V
(
f
1
,...,f
m
)for
V
(
S
). An
affine algebraic set
is a set
n
where
S
X
=
V
(
S
)
⊂ A
⊂ k
[
x
].
