Let be irreducible. Then consider the derivative. is a polynomial of degree n-1 and since , so it is not identically 0. Up to constant factors, the only factors of are and 1, so and are relatively prime, thus is separable, ie has no multiple roots.
Let be irreducible. Then consider the derivative. is a polynomial of degree n-1 and since , so it is not identically 0. Up to constant factors, the only factors of are and 1, so and are relatively prime, thus is separable, ie has no multiple roots.