finite extension and finitely generated extension I don't understand the following statement of Vakil's book (page 107, 3.2.5)

Any ﬁeld extension of $k$ that is ﬁnitely generated as a ring is necessarily also ﬁnitely generated as a module (i.e., is a ﬁnite extension of ﬁelds).

My main question is: what is the definition of "ﬁnitely generated as a ring"?
 A: "Finitely generated as a [structure]" means you can find finitely many elements which produce all of the other elements via repeated application of all of the operations a [structure] has. 
In particular, "finitely generated as a ring" means a ring $R$ has finitely many elements $r_1, \dots r_n$ such that every element of $R$ can be produced from the $r_i$ by repeated addition, negation, and multiplication. Equivalently, this means that every element of $R$ can be expressed as a polynomial in the $r_i$ with coefficients in $\mathbb{Z}$. Yet another equivalent statement is that $R$ is a quotient of a polynomial ring $\mathbb{Z}[x_1, \dots x_n]$. 
But this is the wrong condition for the statement Vakil wants: he needs "finitely generated as a $k$-algebra," which means the same thing as above but also allowing scalar multiplication by elements of $k$. Equivalently, this means that every element of $R$ can be expressed as a polynomial in the $r_i$ with coefficients in $k$, and yet another equivalent statement is that $R$ is a quotient of a polynomial ring $k[x_1, \dots x_n]$.
So, the claim is that if $L$ is a field extension of $k$ which is a quotient of a polynomial ring $k[x_1, \dots x_n]$, then $L$ must in fact be a finite extension of $k$ (in the sense of being a finite-dimensional vector space over $k$). 
