Example

Consider the polynomial f = Xⁿ+1 in Z/pZ[X]. Its derivative Df is nX^n-1.

If p | n then Df = 0, and gcd(f,Df) = f. The lemma does not apply. In fact, f = (X^n/p+1)^p and so multiple roots occur.

Otherwise, gcd(p,n) = 1, so n is invertible in Z/pZ, whence gcd(f,Df) = gcd(Xⁿ+1,X^n-1) = gcd(1,X^n-1) = 1, and, by the lemma, f has no multiple roots.