Transcendental over a field of transcendental

Question

Say I have a field $F$, and $a,b$ are transcendental over $F$. Can I say that $a$ is algebraic over $F(b)$?

Edit. What if $b$ is a monic polynomial in terms of $a$? Would that guarantee that $a$ is algebraic over $F(b)$?

Answer 1

Take the field $\Bbb Q$ and as transcendental numbers $\pi$ and $2\pi$ over $\Bbb Q$. Then $2\pi$ is algebraic over $\Bbb Q(\pi)$ as you can take the linear polynomial $x - 2\pi\in\Bbb Q(\pi)[x]$ with zero $2\pi$.