I'll offer two punchlines for Galois Theory.

1. There's a one-to-one, order-reversing correspondence between intermediate fields of a finite, normal, separable extension $K$ of $F$ and subgroups of the group of automorphisms of $K$ fixing $F$.

2. A polynomial is solvable in radicals if and only if the Galois group of its splitting field is a solvable group.

I'll offer two punchlines for Galois Theory.

1. There's a one-to-one, order-reversing correspondence between intermediate fields of a finite, normal, separable extension $K$ of $F$, and subgroups of the group of automorphisms of $K$ fixing $F$.

2. A polynomial is solvable in radicals if and only if the Galois group of its splitting field is a solvable group.