Galois closure of an extension (reviewed)

If $K$ is a separable algebraic extension of a field $F$, then its Galois closure is the smallest extension field, in terms of inclusion, which contains $K$ and is Galois over $F$. If $K=F(\alpha)$ where $\alpha$ has irreducible polynomial $f$ over $F$, then the Galois closure of $K$ is the splitting field of $f$ over $F$.