Let $R = \mathbb{C}[x_1,…,x_n]/I$ be a quotient of a polynomial ring over $\mathbb{C}$, and let $M$ be a maximal ideal of $R$. How do I show that quotient ring $R/M$ is isomorphic to $\mathbb{C}$?
So I use the fact that $M$ is a maximal ideal of $R$ if and only if $R/M$ is a field. Obviously $\mathbb{C}$ is a field.
How would I use this theorem? Hilbert's Nullstellensatz: the maximal ideals of the polynomial ring $\mathbb{C}[x_1,…x_n]$ are in bijective correspondence with points of complex n-dimensional plane. A point a in $\mathbb{C^n}$ corresponds to the kernel of a substitution map which sends f(x) in $\mathbb{C}[x_1,…x_n]$ to f(a). the kernel of this map is the ideal generated by linear polynomials with roots consisting of the components of a
Best Answer
This follows from Zariski's lemma, and from the fact that $\mathbf C$ is algebraically closed.
Edit: Since your edit, I see that the intended solution was most likely the one provided by Sammy, below. Nevertheless, I think that this proof is the "correct" proof, because it is "independent of coordinates".