Field Theory and Galois

If $E_1,E_2$ are subfields of a larger field $F$, with finite degrees $[F:E_1],[F:E_2]<\infty$, must the intersection $E_1\cap E_2$ have finite degree in $E_1$? In other words, must $[E_1:E_1\cap E_2]<\infty$ hold?