Is a root of $x^4+x^2+1$ constructible?

Question

My problem is as the title, what is that mean that the root of this is constructible or not? Here is what I try:

Let $ u = x^2$ transform the polynomial to $u^2+u+1$ then we know the roots will be $\frac{-1\pm\sqrt3i}{2}$, so we need to check if $Q[1,\sqrt3i]$ is constructible ?

Answer 1

$$x^4+x^2+1=x^4+2x^2+1-x^2=(x^2+x+1)(x^2-x+1).$$ All root of quadratic equation is constructible.

Answer 2

$$ x^4 + x^2 + 1 = \left( x^2 + x + 1\right) \left( x^2 - x + 1\right) $$