If $k\subset F\subset E $ is a tower of fields, prove that $$\operatorname{tr\,deg}(E/k) = \operatorname{tr\,deg}(E/F) + \operatorname{tr\,deg}(F/k).$$
My attempt. Suppose that if $X$ is a transcendence basis of $F/k$ and $Y$ is a transcendence basis of $E/F$. It's easy to show that $X\cup Y$ is algerbaic independent over $k$. But I don't know how to show that it's a transcendence basis. i.e. maximal.