normal closure

Let $K$ be an extension field of $F$. A normal closure of $K/F$ is a field $L\supseteq K$ that is a normal extension of $F$ and is minimal in that respect, i.e. no proper subfield of $L$ containing $K$ is normal over $F$. If $K$ is an algebraic extension of $F$, then a normal closure for $K/F$ exists and is unique up to isomorphism.