Finding the splitting field for $x^4 + x$ over $\mathbb Q$

Question

I've found the roots (with help from wolframalpha): $x = 0$, $x = -1$, $x = $e^i\pi/3$, $x = -(-1)^{2/3}$.

But I don't know how to express it in the form $\mathbb Q(a)$, where $a$ is some real or complex number.

Answer 1

Hint:

As the polynomial factors as $\;x(x+1)(x^2-x+1)$, it's the same as the splitting field of the quadratic polynomial $x^2-x+1$, , which has well-known roots.

Answer 2

The zeros are $0$, $-1$, $\exp(\pi i/3)$ and $\exp(-\pi i/3)$, that is $0$, $-1$, $\zeta$ and $\zeta^{-1}$ for $\zeta=\exp(\pi i/3)$. So they generate the field $\Bbb Q(\zeta)$.