Polynomials: Finding the Splitting Field by Finding the Roots 2

We find the splitting field of x^4+x^2+1 over Q.

[arghyachakraborty]

but I have a question, how to show x^4+x^2+1 is irreducible? it can be factor as x^4+x^2+1=(x^2+x+1)(x^2-x+1)