Kernel of an evaluation homomorphism $\mathbb{C}[X_1,\dots, X_n] \rightarrow \mathbb{C}$ 
Let $R:=\mathbb{C}[X_1,\dots, X_n]$, $a=(a_1,\dots, a_n)\in \mathbb{C}^n$ and $\phi_a:R\rightarrow \mathbb{C}$, $\phi_a(f)=f(a)$. I want to show that $\ker(\phi_a)=(X_1-a_1,\dots, X_n -a_n)$. 

I know that to be true for $n=1$ and I also know that $\ker(\phi_a)$  is a maximal ideal in $R$. Moreover, $(X_1-a_1,\dots, X_n -a_n) \subseteq \ker(\phi_a)$ is obvious.  I am note sure if I can do an induction in this case.  Thank you.

Without loss of generality, you can assume that $a=0 \in \mathbb C^n$. 
Then you want to show that if $f(0,0,\cdots,0)=0$, then $f \in \langle X_1,\cdots, X_n \rangle= \mathfrak m$.
Write $f$ as a sum of monomials, and observe that the constant terms must vanish. This implies that $f \in \mathfrak m$.
