DEFINITION 02.3
Let

be a field.
- We put
and call it (the set of
-valued points of) the projective space.
The class of an element
in
is
denoted by
.
- Let
be homogenious polynomials. Then we put
and call it (the set of
-valued point of) the projective variety
defined by
.
(Note that the condition

does not depend on the choice of the
representative

of
![$ [x]\in \P ^n(\mathbbm{k})$](img21.png)
.)