(a) Let $K$ be a field and let $f(t) \in K[t]$. What does it mean for a field extension $L$ of $K$ to be a splitting field for $f(t)$ over $K$ ?

Show that the splitting field for $f(t)$ over $K$ is unique up to isomorphism.

(b) Find the Galois groups over the rationals $\mathbb{Q}$ for the following polynomials: (i) $t^{4}+2 t+2$. (ii) $t^{5}-t-1$.