Question about algebraic extension of fields

Question

Let $K$, $F$ and $L$ be fields such that $K$ is an algebraic extension of $F$ and $L$ is an algebraic extension of $K$. How do I prove that $L/F$ is algebraic?

Answer 1

Let's show that $\forall \ell \in L$, $\ell$ is algebraic in $F$.

I'm going to use the fact that an element $\ell\in L$ is algebraic if and only if $[F(\ell):F] < \infty$. Since $\ell$ is algebraic in $K$ there exists $p(x) = \kappa_0 + ... + \kappa_nx^n\in K[x]$ such that $p(\ell)=0$. Since $[F(\kappa_0,...,\kappa_n):F] < \infty$ using the tower lemma:$$ [F(\ell):F] = [F(\ell):F(\kappa_0,...,\kappa_n)][F(\kappa_0,...,\kappa_n):F] $$ and that a multiplication of finite numbers is finite (why the first one is finite?), it follows that $[F(\ell):F] < \infty$

Answer 2

Hint: You can put together the following facts:

1. A finite dimensional extension is algebraic. (Proof:: If $L/K$ is a finite dimensional extension, and $\beta \in L$, then there must be some linear dependence over $K$ among $\{ \beta, \beta^2, \beta^3, \ldots \}$.)
2. The tower property for degrees of extensions: $dim_F L = (dim_F K) * (dim_K L)$. (Proof: Write out basis.)