Question about transcendental extension

Question

Let $L=F(u) $, where $u$ is transcendent over $F$ and let $K \neq F$ be a field between $L/F$. How do I show that $u$ is algebraic over $K$?

I tried to create homomorphisms, but it doesn't work.

Answer 1

Pick $v\in K\setminus F$. Since $v\in F(u)\setminus F$, there are two coprime polynomials $p(u),q(u)\in K[u]$ such that $q\ne 0$ and at least one of them is not a constant.

By definition, $K\supseteq F(v)=F\left(\frac{p(u)}{q(u)}\right)$. Now, $u$ is a root of the polynomial $$p(X)-vq(X)\in F(v)[X]\subseteq K[X]$$ the degree of which in $X$ is $\max(\deg p,\deg q)>0$. So $u$ is algebraic over $K$.