P-adic Valuation of Difference of Powers with Coprime Exponent/Proof 2

Proof
We have:
 * $x^n - y^n = \paren {x - y} \paren {x^{n - 1} + \cdots + y^{n - 1} }$

We have to prove that:
 * $p \nmid x^{n - 1} + \cdots + y^{n - 1}$

Let $\map P u = u^n - y^n$.

If $p \divides x^{n - 1} + \cdots + y^{n - 1}$, then $x$ would be a double root of $P$ modulo $p$.

By Double Root of Polynomial is Root of Derivative (or a version of this in modular arithmetic):
 * $p \divides \map {P'} x = n x^{n - 1}$

which is impossible.

Therefore:
 * $p \nmid x^{n - 1} + \cdots + y^{n - 1}$