Transcendental extension - Intermediate fields

Question

A friend and I have been trying to prove the following:

Let $K$ and $E$ be two fields, and let $u$ be transcendental over $K$. If $K\subset E\subseteq K(u)$, then $u$ is algebraic over $E$.

There's no need to give us a solution, we just want to know if the statement is true.

Answer 1

Yes, the statement is true.

Note that $K(u)=K(u,u^2,u^3,.......)$ (why is that so?)

Now, suppose that $x\in E$, also $E$ properly contains $K$, then $x$ is a linear combination of at least one element among $u^i$'s over $K$. So we have, $E$ contains at least one $u^i$.

Now take $f(x)=x^i-u^i$ and you're done.