Ideal generated by $x_1 - a_1, \dots, x_k - a_k$ is prime in polynomial ring over integral domain
I would like to show the ideal $P_k = \langle x_1 - a_1, \dots, x_k - a_k \rangle$ of $R = F[x_1,\dots,x_n]$, where $F$ is a field, $k \leq n$ and $a_i \in F$, is a prime ideal.
In the case where all $a_i = 0$, I think the natural isomorphism $R/P_k \rightarrow F[x_{k+1},\dots,x_n]$ allows us to conclude $R/P_k$ is an integral domain completing the argument. But in the case of general $a_i \neq 0$, I can't see how to set up a similar isomorphism neatly.
Also, am I correct in thinking the result holds if we replace $F$ with any integral domain?
In the case where all $a_i = 0$, I think the natural isomorphism $R/P_k \rightarrow F[x_{k+1},\dots,x_n]$ allows us to conclude $R/P_k$ is an integral domain completing the argument. But in the case of general $a_i \neq 0$, I can't see how to set up a similar isomorphism neatly.
Also, am I correct in thinking the result holds if we replace $F$ with any integral domain?
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