Let $K$ be a field with $a_1,...,a_n \in K $ and $\phi : K[x_1,...,x_n] \rightarrow K , \ \phi (f)=f(a_1,...,a_n) .$
I am trying to show that the kernel of this ring homomorphism is the ideal $ I:= (x_1-a_1,...,x_n-a_n) $. It’s easy to show that $I \subseteq \ker \phi $ but I can’t seem to understand the reverse inclusion. It intuitively makes sense but I can’t formalise it. I have never done a great deal of work in polynomial rings in more than one variable. For one variable it’s easy, because then $f(a)=0 $ if and only if $x-a $ divides $f$ but I don’t know how it really works for polynomials in more variables.
$\phi$ is a surjective map onto a field so $\text{ker}(\phi)$ is a maximal ideal. It contains $I$, which is clearly maximal, hence $I=\text{ker}(\phi)$.