$X^4-5X^2+X+1$ is irreducible in $\mathbb{Q}[X]$

Question

Could anyone advise me on how to efficiently prove $X^4-5X^2+X+1$ is irreducible in $\mathbb{Q}[X] \ ?$

Hints will suffice.

Thank you.

Answer 1

Check the polynomials of degree 1 and 2 over the field Z[3], if they divide the polynomial $x^4+x^2+x+1$. You will find out, that there are none. So, the given polynomial must be irreducible over Q.