This is a basic question about Galois Extension, but I want some details about it.
Let $F$ be a splitting field over $\mathbb Q$ the polynomial $x^8-5\in\mathbb Q[x]$. Recall that $F$ is the subfield of $\mathbb C$ generated by all roots of this polynomial.
(1) Find the degree $[F:\mathbb Q]$ of the number of the number field $F$.
(2) Determine the Galois group $Gal(F/\mathbb Q)$.