Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$. Find examples to illustrate that $[F(a):F(a^3)]$ can be $1,2$ or $3$.
Attempt: $F \subset F(a^3) \subseteq F(a)$
The minimal polynomial for $a^3$ over $F$ is $ x-a^3=0$
I, unfortunately, don't have much idea than this on this problem. Could you please tell me how to move ahead?
Let $K$ be an extension of $F$. Suppose that $E_1$ and $E_2$ are contained in $K$ and are extensions of $F$. If $[E_1:F]$ and $[E_2:F]$ are both prime, show that $E_1 = E_2$ or $E_1 \bigcap E_2 = F $
Attempt: $[K:F] = [K:E_1][E_1:F] = [K:E_2][E_2:F]$
Since, $[E_1:F]$ and $[E_2:F]$ are both prime $\implies [E_2:F]$ divides $[K:E_1]$ and $[E_1:F]$ divides $[K:E_2]$
How do i move ahead?
Thank you for your help.