An intermediate field $K$ is Galois over $k$ iff $\Gal(L/K) \normal \Gal(L/k)$

An intermediate extension is Galois exactly when its fixing subgroup is normal, with quotient the intermediate Galois group.Suggest a slogan

Let $L/k$ be a Galois extension and let $K$ be a field with $k \subseteq K \subseteq L$. Then \[ K/k \text{ is Galois} \iff \Gal(L/K) \normal \Gal(L/k), \] where $\normal$ denotes a normal subgroup, and in that case \[ \Gal(K/k) \cong \Gal(L/k) / \Gal(L/K). \]