Cryptography Reference
In-Depth Information
k
/
k
-
rational points
of
X
Let
k
be an algebraic extension. The
=
V
(
S
)are
k
)
n
(
k
)
n
(
k
):
f
(
P
)
X
(
=
X
∩ A
={
P
∈ A
=
0 for all
f
∈
S
}
.
[
x
], is a
hypersurface
.If
f
(
x
) is a polynomial of
total degree 1 then
V
(
f
)isa
hyperplane
.
An algebraic set
V
(
f
), where
f
∈ k
Informally we often write “the algebraic set
f
=
0” instead of
V
(
f
). For example,
y
2
x
3
). We stress that, as is standard,
V
(
S
) is the set of solutions
over an algebraically closed field.
When an algebraic set is defined as the vanishing of a set of polynomials with coefficients
x
3
instead of
V
(
y
2
=
−
in
” has a different meaning
and the relation between them will be explained in Remark
5.3.7
.
k
then it is called a
k
-algebraic set. The phrase “defined over
k
V
(
x
1
+
x
2
+
2
Example 5.1.2
If
X
=
1)
⊆ A
over
Q
then
X
(
Q
)
= ∅
.Let
k = F
2
and let
V
(
y
8
x
6
y
x
3
2
. Then
X
(
X
=
+
+
+
1)
⊆ A
F
2
)
={
(0
,
1)
,
(1
,
0)
,
(1
,
1)
}
.
±
√
t
):
t
(
t,t
2
):
t
2
,
2
Exercise 5.1.3
Let
k
be
a field. Show that
{
∈ k}⊆A
{
(
t,
∈ k}⊆A
(
t
2
1
,t
3
):
t
2
and
{
+
∈ k}⊆A
are affine algebraic sets.
k
Example 5.1.4
Let
be a field. There is a one-to-one correspondence between the
k
∗
and the
set
k
-rational points
X
(
k
) of the affine algebraic set
X
=
V
(
xy
−
1)
⊂ A
2
.
Multiplication in the field
k
corresponds to the function mult :
X
×
X
→
X
given by
k
∗
corresponds to the func-
mult((
x
1
,y
1
)
,
(
x
2
,y
2
))
=
(
x
1
x
2
,y
1
y
2
). Similarly, inversion in
k
∗
as an algebraic group, which
tion inverse(
x,y
)
=
(
y,x
). Hence, we have represented
we call
G
m
(
k
).
Example 5.1.5
Another elementary example of an algebraic group is the affine algebraic
set
X
=
V
(
x
2
+
y
2
−
⊂ A
2
with the group operation mult((
x
1
,y
1
)
,
(
x
2
,y
2
))
=
(
x
1
x
2
−
1)
y
1
y
2
,x
1
y
2
+
x
2
y
1
). (These formulae are analogous to the angle addition rules for sine and
cosine as, over
, one can identify (
x,y
) with (cos(
θ
)
,
sin(
θ
)).) The reader should verify
that the image of mult is contained in
X
. The identity element is (1
,
0) and the inverse of
(
x,y
)is(
x,
R
y
). One can verify that the axioms of a group are satisfied. This group is
sometimes called the
circle group
.
−
= F
p
(
i
) where
i
2
Exercise 5.1.6
Let
p
≡
3 (mod 4) be prime and define
F
p
2
=−
1. Show
F
p
), where
X
is the circle group from Example
5.1.5
, is isomorphic as a
group to the subgroup
G
that the group
X
(
⊆ F
p
2
of order
p
+
1.
Proposition 5.1.7
Let S
⊆ k
[
x
1
,...,x
n
]
.
1. V
(
S
)
=
V
((
S
))
where
(
S
)
is the
k
[
x
]
-ideal generated by S.
n
where
2. V
(
k
[
x
])
= ∅
and V
(
{
0
}
)
= A
∅
denotes the empty set.
3. If S
1
⊆
S
2
then V
(
S
2
)
⊆
V
(
S
1
)
.
