Galois group of $x^3 + x^2 - 2x - 1$.

Solution 1:

Well I think the point is that $\beta \in \mathbb{Q}(\alpha)$. So since the polynomial is of third degree $E$ will also contain the third root and therefore is the splitting field. It is normal since splitting fields are normal. The degree is three and there is only one group of order three.

Solution 2:

Just for the record, this is the minimal polynomial of $z + z^{-1}$, where $z$ is a primitive $7$-th root of unity. This hint also leads to the result that the Galois group is cyclic of order $3$.

This explains the fact that if $$\alpha = z + z^{-1}$$ is a root, then $$\alpha^{2} - 2 = (z^{2} + z^{-2} + 2) - 2 = z^{2} + (z^{2})^{-1}$$ is also a root.