I would like to see a constructive proof of the existence of maximal ideal in multivariate polynomial ring over $\mathbb C$. I know there are proofs for more general rings using axiom of choice. But I think a constructive proof can give us more insights. The multivariate polynomial ring is a very conrete example, so I believe there should be some conrete proofs (maybe even for the case where the field is not algebraically closed).
To be more precise, suppose there is a proper ideal $I$ of $\mathbb C[x_1, x_2, \cdots,x_n]$, and we know $I$ is generated by finite elements $f_1,f_2,\cdots f_k$. How can we explicitly construct a maximal ideal that contains $I$?
I have found a post here, I don't know whether it asked the same question. In this post the exact form of maximal ideals in $\mathbb C[x_1, x_2, \cdots,x_n]$ is given, I believe it can help (though I don't know whether this result requires the existence of maximal ideals).
Thank you very much if you would like to help.
By Hilbert's Nullstellensatz, we know that maximal ideals of $\mathbb{C}[X_1,...,X_n]$ are exactly the ideals of the form $(X_1-a_1,...,X_n-a_n)$ for $a_1,...,a_n \in \mathbb{C}$.
Suppose $I=\langle f_1,...,f_k \rangle $. Then if we find $a_1,...,a_n$ in $\mathbb{C}$ such that $f_1(a_1,...,a_n)=f_2(a_1,...,a_n)=...=f_k(a_1,...,a_n)=0$, we know by polynomial division that $f_1,...,f_k \in (X_1-a_1,...,X_n-a_n)$ and therefore $I\subset (X_1-a_1,...,X_n-a_n)$ which is maximal.
So all one has to do is find a n-tuple $(a_1,...,a_n)$ in $\mathbb{C}^n$ that is a root of all the $f_i$.