Finite Field Extensions

Question

I'm not terribly familiar with field theory, so I'd appreciate some help understanding this homework problem. For context, this problem is for an algebraic geometry course, in relation to function fields of varieties.

Let $k\subseteq L$ be a field extension. Let $S, R\subset L$ be two subsets. Show that if $k(S)$ is finite over $k$ and $k(R)$ is finite over $k$, then $k(S,R)$ is finite over $k$.

My question is: what exactly does it mean that $k(blah)$ is finite over $k$? And is this equivalent to having transcendence degree of $0$?

Answer 1

Ok so every field is a vector space over itself. Also given an extension of fields $L/K$, the field $L$ is a $K$-vector space (check this).

When the dimension is finite nice things happen so we study that first. The phrase "$L$ is finite over $K$" is referring to this behaviour.

So from your question you are being told that $K(R)$ is finite over $K$, thus there is a finite basis for $K(R)$ as a $K$-vector space. Similarly for $K(S)$. Now how might you make a basis for $K(R,S)$? Can you at least make a finite spanning set?

Answer 2

Being finite means it has finite dimension as a vector space over $k$.

This implies that it is algebraic over $k$ but not every algebraic extension is finite and so, no, it's not the same as having transcendence degree zero.