This question follows Field theory by Steven Roman, Chpater 9, Exercise 20.

Denote the algebraic closure of the finite field $F_q$ by $\Gamma(q)$, and let $a_n$ be any strictly increasing infinite sequence of positive integers. The exercise wants us to prove that $\Gamma(q)=\bigcup_{n=0}^{\infty}GF(q^{a_n})$.

However, if $a_n$ is an arbitrary sequence, we are even unable to prove $\bigcup_{n=0}^{\infty}GF(q^{a_n})$ is a field. I wonder whether the exercise has omitted some condition since the equality doesn't hold under the stated conditions.

In fact, I believe that to demand that $a_n$ is any sequence of positive integers such that any positive integer $k$ divides some $a_n$ is both sufficient and necessary, though I'm not sure.

Hope for answers!