are the integers modulo 4 a field? Basically are the integers mod 4 a field? I want to know because I am reading a text and it has a problem assuming the integers modulo any number are a field

If $q=p^n$, then $\mathbf F_q$ denotes the field with $q$ elements.
You have to know that for any integer $n\ge 1$, there exists a finite field with $p^n$  elements, and this field is unique up to an isomorphism.
It is even unique in the still more restrictive sense: a field with $p^n$ elements is unique within a given algebraic closure of the prime field $\mathbf F_p$. Furthermore, for any two such finite fields,
$$\mathbf F_{p^m}\subseteq\mathbf F_{p^n}\iff m\mid n.$$
