Let $L / K$ be a field extension. State what it means for an element $x \in L$ to be algebraic over $K$. Show that $x$ is algebraic over $K$ if and only if the field $K(x)$ is finite dimensional as a vector space over $K$.

State what it means for a field extension $L / K$ to be algebraic. Show that, if $M / L$ is algebraic and $L / K$ is algebraic, then $M / K$ is algebraic.