Let $L = F(S)$ for $S \subseteq L$. Does there exists a transcendental basis $T \subseteq S$ with $F(S) = F(T)$?

Question

Let $L/F$ be a field extension with $L = F(S)$ for a subset $S \subseteq L$. My first question is:

• Is there a subset $T \subseteq S$ such that $T$ is a transcendental basis?

Now let's say we have $I \subseteq S$ such that $I$ is algebraic independent. My second question is then:

• Is there a subset $T$ with $I \subseteq T \subseteq S$ such that $T$ is a transcendental basis?