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

Proof
We have:
 * $x^n - y^n = \left({x - y}\right) \left({x^{n - 1} + \cdots + y^{n - 1} }\right)$

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

Let $P \left({u}\right) = u^n - y^n$.

If $p \mid x^{n - 1} + \cdots + y^{n - 1}$, 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):
 * $p \mathrel \backslash P' \left({x}\right) = n x^{n - 1}$

which is impossible.

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