In [Field theory by Steven Roman](https://link.springer.com/book/10.1007%2F0-387-27678-5) Chpater 9 Exercise 20, if we write the algebraic closure of finite field $F_q$ as $\Gamma(q)$ and $a_n$ be any strictly incresing infinite sequence of positive integers, the exercise wants 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 current conditions offered.

In fact, I believe the condition 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!