Proof verification on Field Extension

Question

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 $E.$

Proof. Since the degree of the transcendental extension is infinite, $K(u)=K(u,u^2,u^3,.......).$

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

Now consider $f(x)=x^i-u^i.$ This shows that $u$ is a algebraic over $E.$

Is this proof correct? Is it necessary to show that $\{u,u^2,u^3,.......\}$ is a maximal algebraically independent set? Is there anything missing in this proof?

Thanks.

Answer 1

Let us say that $a=\frac{u^2+1}{u}\in E$ for instance. Then $u$ satisfies the equation $$ua = u^2+1$$ which shows $u$ is algebraic over $E$.

I'll let you write the proof for a general case.