If $L_{1}$ and $L_{2}$ are field extensions of $F$ that are contained in a common field, show that $L_{1}L_{2}$ is a finite extension of $F$ if and only if both $L_{1}$
and $L_{2}$ are finite extensions of $F$. (Patrick Morandi, Field and Galois Theory, Exercise $14$ page $14$.)
Can anyone tell me a hint to prove it?
Best Answer
For a hint, you might start out by looking at $L_1(\lambda)$ where $\lambda\in L_2$, and showing that this field is finite over $F$. From there it shouldn’t be too hard.
EDIT, in response to request for further help:
First step, $L_1(\lambda)$ is finite over $L_1$ because $\lambda$ is root of an $F$-polynomial, and thus of an $L_1$-polynomial. Second, this shows that if $\{a_a,a_2,\cdots,a_n\}$ is an $F$-basis of $L_2$, then you have the finite sequence of finite extensions $$ L_1\subset L_1(a_1)\subset L_1(a_1,a_2)\subset\cdots\subset L_1L_2\,. $$