It is well known that if $(A,\mathfrak{m})$ a noetherian local ring, then $\dim{A}$ is finite and $$\dim{A}=\deg{\lambda_A}=\delta_A, $$ where $\lambda_A$ is the Hilbert-Samuel polynomial of $A$ and $\delta_A$ is the minimum number of elements in the system of parameters of $A$.
My question is: It is possible to do an easier proof, using the above theorem, of the fact that $$\dim k[x_1,\dots, x_n] = n $$ when $k$ is a field?