Terminology from Lang, Chapter VIII. 1.

$k\subset K$ field extension and $S$ transcendence base, then $K$ is algebraic over $k(S)$. Why is that so? Lang states as it is something trivial.

Surely we must exploit maximality of $S$ somewhere. I took an element $\alpha\in K$. If it is not algebraic, then it is transcendental over $k(S)$ which means it's also transcendental over $k$. So we can maybe create $S' = S\cup \{\alpha\}$ and show that it is algebraically independent over $k$, but how?

Also, is it always true then: adding transcendental element to an algebraically independent set keeps it algebraically independent?