Given a field $F$ and $R=F[x_1,...x_n]$, show the ideal $(x_1,x_2...x_i)$ is a prime ideal for each i

Question

Given a field $F$ and $R=F[x_1,...x_n]$, show the ideal $(x_1,x_2...x_i)$ is a prime ideal for each i

I think the simplest way of going about this would be to show that the quotient ring $R$ $mod (x_1,...x_i)$ is an integral domain. Is this a correct belief? Modding out by these $i$ generators would effectively reduce all terms of elements in R that contains an $x_i$ to zero, and thus reduce $R$ to $F[x_{i+1},...x_n]$, which is still an integral domain because... for some reason.. right?

Am I on the right track? Is there a better way to go about this?