Quadratic equations over finite fields and class numbers of real quadratic fields

Let p be an odd prime and F-p the finite field with p elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space F-p(n) and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field Q(root p) (where p = 1 (mod 4)) over the field Q of rational numbers can be expressed by means of these formulas.