@BalarkaSen I'm guessing it's not going to be ideal, as I want to show this isn't a PID. So if I will assume it is and try to reach a contradiction.
As $(x_1, \cdots , x_n)$, there is some polynomial $f$ such that $f(x_1, \cdots , x_n) = x_1g_1(x_1, \cdots , x_n) + \cdots + x_ng_n(x_1, \cdots , x_n)$