Solving quadratic equations in many variables

Abstract

Fields are number systems in which every linear equation has a solution, such as the set of all rational numbers $\mathbb{Q}$ or the set of all real numbers $\mathbb{R}$. All fields have the same properties in relation with systems of linear equations, but quadratic equations behave differently from field to field. Is there a field in which every quadratic equation in five variables has a solution, but some quadratic equation in four variables has no solution? The answer is in this snapshot.