Problem 6. (a) Let K be a splitting field of a quintic polynomial $f(x)$ (degree 5) in $\mathbb{Q}[x]$.
Suppose that $f(x)$ has distinct five roots $\alpha, \beta, \gamma, \delta, \eta$, and $[K : \mathbb{Q}] = 5! = 120$.
(1) Find Gal(K/Q).
(2) Find Gal(Q($\alpha, \beta, \gamma, \delta$)/Q($\alpha$)).
(b) Let K/F be a finite Galois extension and let $E_1, E_2$ be intermediate subfields of K/F.
Prove that
Gal(K/$E_1E_2$) $\cong$ Gal(K/$E_1$) $\cap$ Gal(K/$E_2$).
