Let $K\subset L$ be a field extension.
Are the following equivalent?
(1) $L/K$ is purely transcendental. Namely, there is a subset $S\subset L$ such that $S$ is algebraically independent over $K$ and $L=K(S)$.
(2) For all $x\in L\setminus K$, $x$ is not algebraic over $K$.
Please tell me relationship between (1) and (2).
Assume that $L/K$ is purely transcendental with $L=K(S)$ for some algebraically independent set $S$. Let $x\in L$ be algebraic over $K$, i.e. $$x^n+b_1x^{n-1}+\dots+b_n=0$$ for some $b_i\in K$. Now if any element of $S$ would appear non-trivially in $x$ then after clearing denominators in the equation above we would get a non-trivial polynomial equation involving elements of $S$ which is impossible as they were assumed to be algebraically independent over $K$. Hence $x\in K$. This proves (1)$\implies$ (2).
The other implication is not true, let for example $K$ be an algebraically closed field. Then any field extension $L/K$ satisfies (2), but there are field extensions which are not purely transcendental (which is in general actually not that trivial to show, see here for an example).
