# Zeta functions. No.4

Yoshifumi Tsuchimoto

PROPOSITION 4.1   Let be an irreducible polynomial in one variable of degree . Let us consider , an equation in one variable. Then:

DEFINITION 4.2   Let be a ring. A polynomial is said to be homogenius of degree if an equality

holds as a polynomial in variables .

DEFINITION 4.3   Let be a field.
1. We put

and call it (the set of -valued points of) the projective space. The class of an element in is denoted by .
2. Let be homogenious polynomials. Then we set

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 .)

LEMMA 4.4   We have the following picture of .
1. That means, is divided into two pieces a nd .
2. That means, is covered by three open sets'' . Each of them is isomorphic to the plane (that is, the affine space of dimension 2).