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.