Definition. Galois closure [22019]

Let $K / F$ be a separable extension. A field extension $L / K$ is called the Galois closure of $K$ over $F$ if

1. $L / F$ is Galois, and
2. is minimal in the sense that $K \subseteq E \subseteq L$ and $E / K$ is Galois $\implies$ $E = L$.