Hey I want to check if my solutions for this problem are right:
Show that the polynomial $f := x^4 + x^3 + x^2 + x + 1 ∈ \mathbb{Q}[X]$ is irreducible and determine the splitting field $E$ of $f$ and the Galois group $Gal(E/\mathbb{Q})$
Since $f$ Cyclotomic polynomial it is irreducible. $E=\mathbb{Q}(e^{\frac{2\pi i}{5}}=: \xi)$
The degree of the extension is $[E:\mathbb{Q}]=4$, right?
So we have $[E:\mathbb{Q}]=4=Gal(E/\mathbb{Q})$
And $Gal(E/\mathbb{Q})=\{\tau_1,...,\tau_4\}$
Where $\tau_i: \xi \rightarrow \xi^i$ and therefore is $Gal(E/\mathbb{Q})$ isomorphic to $\mathbb{Z}_4$