Intersection of function fields

Question

Let $k_0 \subset K$ be a field extension and let $k_0 \subset k_1, k_2 \subset K$ be subextensions.

Is it true that $$k_1(t) \cap k_2(t) = (k_1 \cap k_2)(t)$$ ? We clearly have $\supseteq$, but if we have $$\frac{P_1(t)}{Q_1(t)} = \frac{P_2(t)}{Q_2(t)} \in k_1(t) \cap k_2(t),$$ with $P_i, Q_i \in k_i(t)$, it does not mean that $P_i, Q_i \in (k_1 \cap k_2)(t)$.

Answer 1

Assume $P_i$ is coprime with $Q_i$ for $i=1,2$ and $Q_1,Q_2$ are monic, then $\frac{P_1(t)}{Q_1(t)} = \frac{P_2(t)}{Q_2(t)}$ implies $P_1=P_2$ and $Q_1=Q_2$.

Hence, $P_1 \in k_1[t] \cap k_2[t] = (k_1 \cap k_2)[t]$, likewise for $Q_1$. Whence $\frac{P_1}{Q_1}\in (k_1 \cap k_2)(t)$.