so as it is in the question.
Clearly $F(a^3)$ is a subfield of $F(a)$. So we know that $[F(a):F(a^3)][F(a^3):F]=5$ the problem now is, how do we know if $[F(a):F(a^3)]=5$ or $[F(a^3):F]=5$. The trouble is that $F(a^3)$ could possibly be $F(a)$ or $F$ depending on the multiplicative order of $a$. Or so I think, am I making a mistake somewhere? The question then goes on to ask if the argument applies if we change $a^3$ to $a^2$ or $a^4$. Any hints would be appriciated.
As you stated, $$5 = [F(a):F] = [F(a):F(a^3)][F(a^3):F].$$ If the second term in the product is 1, then $a^3 \in F$, so that $[F(a):F] \le 3$. We know that's false! Hence the second value is 5, since it's not 1, and 5 is a prime.