Transcendence degree of fraction field

Question

Let $k$ be a field and $p \in k[x_1, \dots, x_n]$ an irreducible element. Is there an elementary way to prove that $\operatorname{tr.deg}_k \mbox{Frac}(k[x_1, \dots, x_n]/(p)) = n-1$?

Answer 1

Pick one variable say $x_n$ which occurs in $p$ then consider the quotient as an algebraic extension of a field of transcendence degree $n-1$.