Prime ideal in $\mathbb{C}[X_1,\dots,X_n]$

Question

Let $f_i\in\mathbb{C}[X_i]$ be a monic polynomial of $X_i$, and we consider : $$ I=(f_1,f_2,\dots,f_n) \in \mathbb{C}[X_1,X_2,\dots,X_n] $$ the ideal formed by $f_i$. I think that $I$ will be maximal iff each $f_i$ is of degree $1$, but when this ideal will be prime and when it will be trivial ?

Answer 1

Over $\mathbf{C}$ or any other algebraically closed feel $K$, you proceed as follows. Suppose $I$ is a maximal ideal of $K[x_1,\dots , x_n]$. Then $K[x_1,\dots,x_n]/I$ is a field finitely generated as an algebra over $K$. By Zariski's lemma, it is a finite (hence algebraic) extension of $K$.

The hypothesis $K$ algebraically closed now implies that $K[x_1, \dots, x_n]/I$ is itself the copy of $K$ in this quotient. Therefore, if the class of $x_i$ in the quotient equals the class of $\alpha_i \in K$, then $x_i - \alpha_i$ must be in the ideal $I$.

Since the ideal $(x_1 - \alpha_1,\dots , x_n - \alpha_n)$ is maximal, you conclude that $I = (x_1 - \alpha_1,\dots , x_n - \alpha_n)$.

(Can you see what goes wrong when $K$ is not algebraically closed?)

(And I recommend that you use the case $n = 2$ as an exercise to investigate what prime but not maximal ideals of $\mathbf{C}[x, y]$ look like)