MHB Is x^4+x^3+1 Irreducible in Q[x] and Z[x]?

[Joe20]

Hi all, appreciate your help to look through my answers to see if they are correct.

Thank you.

 

[Opalg]

Looks good to me. If you wanted to avoid the long division, you could observe that the only candidate for a quadratic divisor of $\overline{f}(x)$ is $x^2+x+1$ (since you have already ruled out the other three quadratic polynomials). So the only possible factorisation would be if $x^4+x^3 + 1 = (x^2+x+1)^2$. But $(x^2+x+1)^2 = x^4+x^2 + 1 \ne x^4+x^3 + 1$. It follows that $x^4+x^3 + 1$ is irreducible.