I want to prove or disprove the following:
Let $L_1$ and $L_2$ be two finite extensions of field $k$ inside of an extension $L/k$. Moreover, $L_1 \cap L_2 = k$. Then the degree of the composite field $L_1 L_2$ is $[L_1 L_2 : k] = [L_1 : k] [L_2 : k]$.
I want to solve this problem with basic field theory (I haven't studied Galois theory). Thanks in advance.
Below is what I've done so far.
Let $[L_1 : k] = m$, $[L_2 : k] = n$, and write $L_1$, $L_2$ as $L_1 = k(\alpha_1, \dots, \alpha_m)$, $L_2 = k(\beta_1, \dots, \beta_n)$, where $\alpha_1, \dots, \alpha_m$ is a $k$-basis for $L_1$, and $\beta_1, \dots, \beta_n$ is a $k$-basis for $L_2$. Then we can easily show that
$$[L_1 L_2 : k] \le [L_1 : k] [L_2 : k],$$
where the equality is achieved iff $\beta_1, \dots, \beta_n$ are linearly independent over $L_1$.
I tried to use $L_1 \cap L_2 = k$ to prove the linear independence but failed.