Exercise 15, § 2

Let A be a commutative ring and n an integer ≥2 such that for all a ∈ A the equation nx = a has a solution x ∈ A. Let m be an integer ≥1 and u a polynomial in A[X] of the form X^{mn} + f(X) where f is of degree ≤ mn − 1; show that there exists a polynomial v ∈ A[X] of the form X^m + w, where w is of degree ≤ m − 1, such that u − v^n is zero or of degree < m(n − 1).