Maximal cyclotomic extension of a finite field
= Maximal cyclotomic extension of a finite field
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.