This is from Reid's Undergraduate Commutative algebra.
Let $a=(a_1,\dots,a_n\in K^n$ where $K$ is an algebraic field extension of $k$. Determine the image and kernel of the evaluation map $e_a:k[x_1,\dots,x_n]\to K$ defined by $e_a(f)=f(a_1,\dots,a_n)$, and prove that $(x_1-a_1,\dots,x_n-a_n)\cap k[x_1,\dots,x_n]$ is a maximal ideal in $k[x_1,\dots,x_n]$ (The intersection is in $K[x_1\dots,x_n]$.
I really don't know how to tackle the question. My intuition is that the image has to be the "small" field $k$, since at most algebraic roots are added, but I don't know what to say for the kernel and the ideal.