Is $x^3+y^3+z^3-1$ irreducible over a field $k$ of characteristic $\neq 3$?

Is $x^3+y^3+z^3-1$ irreducible over a field $k$ of characteristic $\neq 3$?

If I can show that $y^3+z^3-1$ is irreducible, I believe I can use Eisenstein's criterion. But I don't see how to show even this.

The other approach on my mind is to show that $k[x,y,z]/(x^3+y^3+z^3-1)$ is an integral domain, but this doesn't seem particularly promising either.

Solution 1:

To show that $y^3+z^3-1$ is irreducible, you may try to show that $z^3-1$ has no repeated prime factor. This is where the restriction on the characteristic comes into play. Then, you can use Eisenstein's Criterion.