Let $k$ be a field and $R$ be the polynomial ring $k[x_{1},...,x_{n}]$.
$\mathbf{Problem}$: Show that any nonzero prime ideal of $R$ contains an element $f$ such that $R(f)$ is transcendental over $R$ with transcendence degree $1$. Use induction to show that the Krull dimension of R is at most $n$.
First of all, I'm confused if there is a typo and it should say "$k(f)$ is transcendental over $k$". In any case, the most I've got towards a solution is observing that any nonzero element of a prime ideal of $R$ generates an extenstion of $k$ of transcendence degree $1$, but that doesn't seem very helpful for the induction.
EDIT: It is a typo, it should be $k$ instead of $R$. The whole point of this exercise is to reduce to the case where the algebra is a polynomial ring. But it seems like to do the induction you need to pass to a quotient which might not be a polynomial ring.