Exercise 21, § 1

Let $f$ and $g$ be two relatively prime polynomials (with complex coefficients), the degree of $f$ being strictly less than that of $g$. Let $p$ be the gcd of $g$ and its derivative $g'$, and let $q$ be the quotient of $g$ by $p$; show that there are two uniquely determined polynomials $u,\ v$ whose degrees are, respectively, strictly less than those of $p$ and $q$, such that

Deduce that the coefficients of $u$ and $v$ belong to the smallest field (over $\mathbf{Q}$) containing the coefficients of $f$ and $g$ and contained in $\mathbf{C}$.