Showing that the polynomial $x^n+y^n-1$ is irreducible over $\mathbb{C}[x,y]$

Question

I’d like to show that the polynomial $x^n+y^n-1$ ($n \geq 1$) is irreducible over $\mathbb{C}[x,y]$ (because I want to see that $x^n+y^n=1$ defines a smooth curve). I’ve been told this can be done by using (a generalization of?) Eisenstein’s criterion for $\mathbb{C}[x][y]$, since $\mathbb{C}[x]$ is an UFD, but I’m not sure how to proceed since I only know how to apply it over the rational numbers. Any help or hints would be welcome.

Answer 1

Hint: Write the polynomial as $y^n+(x-1)(x^{n-1}+x^{n-2}+...+x+1)\in\mathbb{C}[x][y]$.