Maximal cyclotomic extension of the rational numbers
= Maximal cyclotomic extension of the rational numbers
The maximal cyclotomic extension of $\mathbb Q$ is a proper subfield of $\overline{\mathbb Q}$. Every finite subextension inside a cyclotomic field is Galois and abelian, whereas $\mathbb Q(\sqrt[3]{2})/\mathbb Q$ is not normal.