If $u$ is transcendental over a field $F$, can it be the case that $F(u) = F(u^2)$?

Question

As seen here, if $u$ is algebraic over a field $F$ and $[F(u):F]$ is odd then then $F(u) = F(u^2)$. If instead $u$ is transcendental over $F$, can it be the case that $F(u) = F(u^2)$?

Answer 1

If $F(u) = F(u^2)$, then you can write $u$ as a rational function in $u^2$. Say $u = f(u^2)/g(u^2)$ for some polynomials $f$ and $g$. But then $ug(u^2) - f(u^2) = 0$, implying that $u$ is algebraic over $F$.