I'm reading up on a proof of Hilbert's Nullstellensatz which uses the Artin-Tate lemma. I followed all of it except for one step, which is probably quite elementary, but my brain may be too fried from the rest of the proof to find the logic.
Let $k$ be an algebraically closed field, and $I$ be a maximal ideal of the polynomial ring $k[x_1,...,x_n]$. After much proof, we have that $k[x_1,...,x_n]/I=k$. Then for each $x_i$, there exists $a_i\in k$ such that $x_i-a_i\in I$.
(Finishing the proof from here, $I$ must contain the ideal $(x_1-a_1,...,x_n-a_n)$, and that ideal is maximal, so $I$ is exactly that.) What's the reason for why such an $a_i$ exists? I appreciate your help!