Maximal cyclotomic extension

For a field $K$ inside a fixed algebraic closure, the maximal cyclotomic extension is the union of the fields $K({\zeta }_n)$ generated by roots of unity.

The maximal cyclotomic extension of $\mathbb{R}$ is $\mathbb{C}$, since adjoining a primitive fourth root of unity adjoins $i$.

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.

The maximal cyclotomic extension of ${\mathbb{F}}_p$ is ${\overline{\mathbb{F}}}_p$. A primitive $(p^m-1)$th root of unity generates ${\mathbb{F}}_{p^m}$ over ${\mathbb{F}}_p$, and these finite fields exhaust the algebraic closure.