# Why is the kernel of $k[x_1,\dots,x_n]\to k$ a maximal ideal?

In Reid's Undergraduate Commutative Algebra, $k$ a field and a point $P=(a_1,\dots,a_n)\in k^n$ determine a homomorphism on the the polynomial ring of functions $k[x_1,\dots,x_n]\to k$ by $g\mapsto g(P)$ for $g\in k[x_1,\dots,x_n]$.

The kernel of this homomorphism is $(x_1-a_1,\dots,x_n-a_n)$. Reid denotes this as $m_P$ and says it is maximal. Why is it maximal though? I know $k[x_1,\dots,x_n]/(x_1-a_1,\dots,x_n-a_n)$ is isomorphic to the image of the map in $k$, which is a commutative subring of a field. Is there a way to see it is a field, to get that the kernel is maximal? I figure if $g(P)\neq 0$ in $k$, then is $1/g(P)$ the value of some other polynomial $f(P)$?

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The map $k[x_1, \ldots, x_n] \to k$ that you wrote down is surjective, since for each $b \in k$ we can always take $g$ to be the constant polynomial $g(x_1, \ldots, x_n) = b$, for which $g(P) = b$.