# evaluation maps on algebraic field extensions

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.

## 1 Answer

The image contains each $a_i$ since $e_a(x_j)=a_j$. The $a_j$ need not lie in $k$, so the image need not be $k$.

The image is the ring generated over $k$ by the $a_j$, and this is a field, since the $a_j$ are algebraic over $k$.

The kernel is the intersection $k[x_1,\ldots,x_n]\cap(x_1-a_1,\ldots,x_n-a_n)$ that you identify.