Corollary 2

Let $K$ be a field, $A$ a subring of $K$ and $L$ the field of fractions of $A$. If $K$ is a finitely generated $A$-algebra, $[K:L]$ is finite and there exists $a \neq 0$ in $A$ such that $L = A[a^{-1}]$.

It follows from Corollary 1 that there exist elements $x_1, \ldots, x_n$ of $K$ and an element $a \neq 0$ of $A$ such that $x_1, \ldots, x_n$ are algebraically independent over $A$ (and therefore over L) and that K is integral over the subring $A[x_1, \ldots, x_n, a^{-1}]$. Then it follows from §2, no.1, Lemma 2 that $A[x_1, \ldots, x_n, a^{-1}]$ is a field. But the only invertible elements of a polynomial ring $C[Y_1, \ldots, Y_n]$ over an integral domain C are the invertible elements of C; applying this remark to $C = A[a^{-1}]$, it is seen that necessarily $n = 0$ and that $A[a^{-1}]$ is a field equal to L by definition of the latter. As K is integral over L and is a finitely generated L-algebra, the degree [K:L] is finite (§ 1, no. 1, Proposition 4).