# Write the algebra closure of \$ F_p \$ as a union of finite fields

In the field theory of Steven Roman Chpater 9 Exercise 20, if we write the algebraic closure of the finite field $$F_q$$ as $$Gamma (q)$$ Y $$a_n$$ any infinite sequence of positive integers that is strictly increased, the exercise wants to prove that $$Gamma (q) = bigcup_ {n = 0} ^ { infty} GF (q ^ {a_n})$$.

However, if $$a_n$$ It is an arbitrary sequence, we are even unable to prove $$bigcup_ {n = 0} ^ { infty} GF (q ^ {a_n})$$ it is a field I wonder if the exercise has omitted any condition since the equality is not fulfilled under the current conditions offered.

In fact, I think the condition that $$a_n$$ is any sequence of positive integers so that any positive integer $$k$$ divide some $$a_n$$ It is sufficient and necessary, although I am not sure.