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

Letp be an odd prime and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{F}_p $$ \end{document} the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{F}_p^n $$ \end{document} and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Q}(\sqrt p )$$ \end{document} (wherep≡1 (mod 4)) over the field ℚ of rational numbers can be expressed by means of these formulas.