This has been asked before at least two different times, but I am still confused so I thought of asking it myself.
The corollary goes like this
Let $k$ be a field, $B$ afinitely generated $k$-algebra. If $B$ is a field, then it is a finite algebraic extension of $k$.
$B$ being a finitely generated $k$-algebra means than $B=k[b_1,...,b_n]$ for some $b_i \in B$. B being finite algebraic finite extension means that those $b_i$ are algebraic over k. Therefore the corollary says that if $B$ is a field then $b_i$ are algebraic. I do understand the proof using all the previous propositions/corollaries.
But why isn't it obvious? If some of the $b_i$ are transcendental, isn't it obvious that $B$ is not a field, as it is a polynomial ring? To be more precise, say $B=k[b]$. If b is transcendental then $B=k[X]$ (not a field). Similarly if $B=k[b_1,...,b_n, t_1,...,t_m]$ where $b_i$ are algebraic and $t_i$ transcendental, then let $C=k[b_1,...,b_n]$ and thus $B=C[t_1,...,t_m]=C[x_1,...,x_m]$. $C$ is a field and thus $B$ is a polynomial ring (not a field).
I suppose I am missing something really obvious, so I 'd appreciate any help in clarifying it.