Maximal ideals of polynomial ring We know that if $k$ is algebraically closed, then each maximal ideals of $k[x_1, x_2, \ldots , x_n]$ are  of the form $(x_1 - a_1, x_2 - a_2, \ldots, x_n - a_n),$ where $a_1, a_2, \ldots , a_n \in k$ (Hilbert's Nullstellensatz Theorem). In the case when $k$ is not algebraically closed is it correct to say that a maximal ideal $m$ of  $k[x_1, x_2, \ldots, x_n]$ has residue field $k$ if and only if $m = (x_1 - a_1, x_2 - a_2, \ldots, x_n - a_n)$ for some $a_1, a_2, \ldots, a_n \in k.$
Thank you.

Yes. Set $R=k[x_1...x_n]$ and suppose $R/m\cong k$. Then we have a homomorphism $R\to k$ given by composing $R\to R/m$ with this isomorphism, and each $x_i$ is sent to some element $a_i$ of $k$. This completely determines the kernel of $R\to k$, for instance since $R$ is a free algebra, and so realizes $m$ in the desired form.
