Let $R_n = \mathbb{Z}[x_1,x_2, \ldots,x_n]$ be a $n$ variable polynomial ring. Given a maximal ideal $I$ of $R_n$, I want to find about the characteristic of the field $R_n/I.$
If $n=1$, it is well known that the maximal ideals are in the form of $(p, f(x))$ where $f(x)$ mod $p$ is irreducible in $\mathbb{Z}_p[x].$ So $R_n/I$ will have characteristic $p$ for some prime $p$.
We may see that $R_n$ is a finitely generated $\mathbb Z$-algebra, and field of char $0$ contains rational numbers which is not finitely generated $\mathbb Z$-algebra, so it follows that $R_n$ has non zero characteristic.
I do not want to use the approach of finitely generated algebras, is there any way to deduce the same without using this approach?