I have a splitting field question, but I will try my best at attempting the problem to the best of my ability.
Consider the function $f(x) = x^{10} + 1$. I want to find a primitive element of the complex splitting $K$ field of $f(x)$ over $\mathbb{Q}$.
Attempt: $x^{10} + 1 \Rightarrow x^{10} = -1$. Therefore, $x^{20} = 1$, and the Galois group is $G(\mathbb{Q}(e^{\frac{2 \pi i}{20}}) / \mathbb{Q})$ which is isomorphic to $\mathbb{Z}_{20}^{*}$. Now from here I am not sure where to get the splitting field.
