The quote is followed immediately by this: "It extends naturally to equations with coefficients *in any field*, but this will not be considered in the simple examples below." Emphasis on 'in any field' is mine. Among the other fields that can be considered include the Galois fields, which are another name for finite fields. (There are also infinite fields other than the rationals, so 'in any field' does not just mean Galois fields/finite fields.) https://en.wikipedia.org/wiki/Finite_field

Galois fields have nothing to do with being able to represent rational numbers in a computer: elements of a finite field aren't even rational numbers.