Exercise 28, § 1

a) Let K be an algebraically closed commutative field, let f be a nonconstant monic polynomial in K[X], and let g be a polynomial in K[X, Y]; suppose that for each of the roots α_i of f there exists a root β_i of g(α_i, Y) = 0 such that $\frac{\partial g}{\partial Y}(\alpha_i, \beta_i) \neq 0$. Then show that there exists a polynomial h(X) ∈ K[X] such that g(X, h(X)) ≡ 0 (mod f(X)). (Using Prop. 4, reduce to the case f(X) = X^m (for some m ≥ 1); notice that the ring K[X]/(X^m) is isomorphic to the quotient ring K[[X]]/(X^m), and use the Cor. of IV, p. 37). b) In particular show that if m is an integer not divisible by the characteristic of K, then for every pair of coprime monic polynomials f and f_0 in K[X], there exists a polynomial h(X) in K[X] such that (h(X))^m ≡ f_0(X) (mod f(X)).