Prime Field Extensions Equal or Intersection Base Field

Question

Let E1 have degree p over F and E2 have degree p' over F, where p,p' are prime numbers. Show that either E1=E2 or E1nE2=F (intersection).

Would anyone have any ideas on how to approach this problem/solve it?

Answer 1

The idea is quite simple: the tower law states that if $K \supset L \supset F$ is a tower of field extensions, then $[K:L][L:F]=[K:F]$.

Also, if $K \supset L$ is a field extension and $[K:L]=1$, then $K = L$.

Note that we have two towers here: $E_1 \supset E_1 \cap E_2 \supset F$ and $E_2 \supset E_1 \cap E_2 \supset F$.

The rest is arithmetic. Good luck!