TITLE: Definition of degree of finite morphism plus context
QUESTION [10 upvotes]: Let $f: X \rightarrow Y$ be a finite morphism of schemes, defined here,
http://en.wikipedia.org/wiki/Finite_morphism
I always assumed that the degree of $f$ was the degree of the induced field extension
$$
[K(Y):K(X)],
$$
of course only defined if $f$ is dominant (hence surjective).
However, is this standard?
Or is the degree of $f$ defined as: Let $f$ locally be given by maps of rings $B_i \rightarrow A_i$. The degree of $f$ is the minimal number of elements of $A_i$ that generate it as a $B_i$ module?
I'm almost sure this does not work.
Q1: Why not? (excuse my lack of commutative algebra knowledge) (i am more than happy with just a sketchy answer to this!)
Q2: Does it work if we assume $X$ and $Y$ to be varieties over $\mathbb{C}$? If so/not, why?
Q3: What is indeed the commonly agreed definition of the degree of a finite map?
That's quite a lot, thanks for your answer!

REPLY [3 votes]: Besides Adeel's nice answer, one can find another definition of the degree in case the morphism is flat and finite (and the schemes are locally Noetherian) in the Stacks project 02K9. This is actually the thing I hoped it would be, as I stated in the question.