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First of all, I don't think this is the smartest question and I feel quite ashamed about it, but here goes nothing:
I wanted to prove that in the polynomial ring $k[x_1, \ldots, x_n]$ the ideal $I = (x_1 - a_1, \ldots, x_n - a_n)$ is maximal and tried as follows: I have defined a map
$$\varphi: k[x_1, \ldots, x_n] \to k: f(x_1, \ldots, x_n) \mapsto f(a_1, \ldots, a_n)$$ for which I was able to prove that it is a surjective ring homomorphism. Hence I know from the first isomorphism theorem that
$$k[x_1, \ldots, x_n]/ \ker(\varphi) \cong k$$ showing that $\ker(\varphi)$ is maximal. I see that the ideal $I \subset \ker(\varphi)$ but I am stuck on the other direction: suppose $f \in \ker(\varphi)$, then $f(a_1, \ldots, a_n) = 0$. How do I show that it must be of the form $(x_1-a_1)g_1 + \ldots (x_n - a_n)g_n$ for $g_i \in k[x_1, \ldots, x_n]$?