Nature of the Roots of an Equation with Real Coefficients

So far the roots we have manipulated could, in principle, be any complex numbers. But when an equation's coefficients are constrained -- to be real, or more strongly to be rational -- its non-real and irrational roots turn out to be constrained too: they can never occur singly. This section proves the two theorems that make this precise, and shows how each one is used to recover an equation's re …