I'm trying to solve the problem Sec. 2 - 1.16 in Shafarevich's book Basic Algebraic Geometry, vol. 1, second ed. My attempt is to use the previous exercise Sec. 2 - 1.12. So I only need to show that the polynomial $x_0^4+x_1^4+x_2^4+x_3^4 - a x_0x_1x_2x_3$ is irreducible for every value of $a \in \mathbb{C}$ and compute the common roots of the polynomial and the partial derivatives.
My problem is to show that the polynomial is irreducible. I've found the following question in the MSE site related with the problem:
Does $x_0^4+x_1^4+x_2^4+x_3^4 - a x_1x_2x_3x_4$ really define a surface?
In an answer, a user says that the irreducibility of the polynomial follows from an expression given there. What is he referring to? I've never seen something similar to this while proving that a polynomial is irreducible.
Thanks in advance!