Nature of Roots and Nature of Coefficients of Polynomial Equations

Vieta's formulae link coefficients to combinations of roots. This section asks a different question: what does the type of the coefficients (real, or more restrictively rational, or more restrictively still integer) force about the type of individual roots? The answer is a family of "roots come in pairs" theorems — imaginary roots pair as complex conjugates (§3.4.1), certain surd roots pair as conjugate surds (§3.4.2) — plus a sharp characterisa …