As the title says, I need to find the primitive element of the splitting field of $1 + x + x^2 - x^5$ over $\mathbb{Q}$.
Firstly, I would proceed by finding the roots as the splitting field has to contain all the roots. So I rewrite the polynomial as: $$-(x^2+1)(x^3-x-1)$$ But here things get awful really quick - apart from the obvious roots $\pm i$, the roots of the cubic are far from being something nice.
Am missing some smart idea on how to avoid such nasty expressions when solving the problem?
UPDATE
The polynomial is a characteristic polynomial of an integer 5x5 matrix with determinant 1 and zero trace. Would some transformation of my matrix possibly help?
Let $\alpha$ be a root of the cubic. The discriminant of the cubic is $-23$ (I think --- better check this), so the splitting field of the cubic is ${\bf Q}(\alpha,\sqrt{-23})$. Then the splitting field of the polynomial is ${\bf Q}(\alpha,\sqrt{-23},i)$. The chances are that $\alpha+\sqrt{-23}+i$ is a primitive element (not the primitive element, there's no such thing). Try to prove that it has degree 12.