 Solution

The Galois group $\mathrm{Gal}(L/K)$ is the group of all field automorphisms of $L$ fixing every element of $K$, with composition as its operation. A finite extension is a Galois extension if it is both normal and separable; equivalently, it is a splitting extension of separable polynomials, or its group has order $[L:K]$. Normality means that every irreducible polynomial over $K$ with a root in $L$ splits there.