Cryptography Reference
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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.
 
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