holds as a polynomial in variables .
and call it (the set of -valued points of) the projective space. The class of an element in is denoted by .
and call it (the set of -valued point of) the projective variety defined by .
That means, is divided into two pieces a nd .
That means, is covered by three ``open sets'' . Each of them is isomorphic to the plane (that is, the affine space of dimension 2).
We quote the famous
where are algebraic integers.
where is an integer.
It is a profound theorem, relating rational points of over finite fields and topology of .
The following proposition (which is a precursor of the above conjecture) is a special case
where is an integer which satisfies .
For a further study we recommend [1, Appendix C],.