Proving $k[x_1, ..., x_n]/J \cong k$ iff $J = (x_1 - a_1, ..., x_n - a_n)$, where J is maximal

Question

Let $J \subseteq k[x_1, ..., x_n]$ be a maximal ideal. Prove that $k[x_1, ..., x_n]/J \cong k$ iff $J = (x_1 - a_1, ..., x_n - a_n)$ for some $a_1, ..., a_n \in k$

I have a few ideas, but unsure of how to proceed. I know that modding out by a maximal idea gives us a field, which gives us some useful properties to consider (including the properties of integral domains).

Answer 1

Let $f:k[x_1, \ldots , x_n]/J\to k$ be an isomorphism, and let $$f(x_i)=a_i$$ then $f(x_i-a_i)=0$ and since $f$ is an isomorphism we have $x_i-a_i\in J$. Now we see $(x_1-a_1,\ldots )\subseteq J$. However since this last ideal is maximal, we have equality.