The question that was left unanswered during a lecture was
For a differential field $K$ and $x$ an element of a differential field extension of $K$, is it necessarily the case that for any element $x$ that is differentially algebraic over $K$, $K\langle x\rangle $ has finite transcendence degree over $K?$ Is the converse true as well?
Definitions I'm working with:
$x$ is differentially algebraic over $K$ if $P(x) = 0$ for some $P \in K\{X\}^{\neq}$
If $S$ is a subset of $K$ which is algebraically independent over $k$, and if the cardinality of $S$ is greatest among all such subsets, then we call this cardinality the transcendence degree of $K$ over $k$.
And $K\langle y \rangle$ is just $K(x, x', x'')$
My intuition is that the answer is yes to both, but I don't see a rigorous way to prove it. Any help would be appreciated.