Irreducibility of $x^4+x^3+x^2+x+1$

Question

How can I see that the polynomial $x^4+x^3+x^2+x+1$ is irreducible over $\mathbb Q$?

I can't apply eisenstein's theorem. What are the other possibilities?

Answer 1

It's the fifth cyclotomic polynomial. All cyclotomic polynomials are irreducible over $\mathbb Q$.

Answer 2

You can actually apply Eisenstein criterion the polynomial $(x+1)^4+(x+1)^3+(x+1)^2+(x+1)+1=x^4+5x^3+10x^2+10x+5$.

This yields irreducibility of the original polynomial.