I want to show that the polynomial given by $f(x,y,z) = y^{n} - zx^{n-1} + z^{n}\in k[x,y,z]$ is irreducible for $n\in\mathbb{Z}_{>1}$, where $k$ is an algebraically closed field.
My idea: was to view $f$ as a polynomial in $k[x,z][y]$, and since $k[x,z]$ is a UFD and since $z$ is irreducible in $k[x,z]$ we see that $f$ is an Eisenstein polynomial. We namely have $z\rvert (-zx^{n-1} + z^{n})$, and $z^{2}\not| (-zx^{n-1} + z^{n})$. And since $f$ is also primitive we find that $f$ has to be irreducible in $k[x,y,z]$.
My (possible) Problem: Is my reasoning correct? I have the feeling that for some $n$ and some characteristics of $k$ this reasoning could go wrong, especially at the part where I say that $z\rvert (-zx^{n-1} + z^{n})$, and $z^{2}\not| (-zx^{n-1} + z^{n})$.
Other Example where Problem with Characteristic appear: $g(x,y,z) = y^{n} - x^{n-1} + 1\in k[x,y]$. If $char(k)\nmid n-1$, then one can see that it is an Eisenstein polynomial with respect to $x-1$, but what happens when $char(k)\mid n-1$? Is it still irreducible then? And how does one show this?