Normal closure

Summary

The normal closure of an algebraic extension $L/K$ is the smallest extension of $L$ that is normal over $K$. It is obtained by adjoining to $L$ all the roots of the minimal polynomials over $K$ of the elements of $L$. It exists and is unique up to an isomorphism over $L$. When $L/K$ is finite and separable, its normal closure is the smallest Galois extension of $K$ containing $L$. This lets Galois theory study extensions that are not themselves Galois.