I am a beginner in algebraic geometry and I have a doubt over the rings/fields involved in algebraic varieties.

Consider this definition:

> 
> An aﬃne algebraic variety over the ﬁeld $\mathbb{F}^n$ is the zero set $V (p\_1,...,p\_k) ⊂ \mathbb{F}^n$ of a ﬁnite number of polynomials $p\_1, ..., p\_k \in \mathbb{F}[x\_1,...,x\_n]$
> 

I realise there are types of varieties other than affine ,but even looking ahead in Wikipedia [here](https://en.wikipedia.org/wiki/Algebraic_variety#Projective_varieties_and_quasi-projective_varieties) my question is this: 

In algebraic geometry, are the polynomial variables (hence the zeros) always required to live in (a cartesian product of) the field of the coefficients ?

For example, do we also study cases where, say, the coefficient ring is $\mathbb{Z\_4}$ and the variables (hence the zeros) live in the field $\mathbb{C}$ or vice-versa ?