This is not correct, there are a finite number of polynomials for any finite field. This is using the usual definition for polynomial equivalence, where two polynomials are equal if they take the same values over all points in the domain.
You are talking about the polynomial ring, which is not a field.
The finite field F2 has four polynomials: 0, 1, x, and x+1. Other polynomials do not exist, because x^2=x.
You are talking about the polynomial ring, which is not a field.
The finite field F2 has four polynomials: 0, 1, x, and x+1. Other polynomials do not exist, because x^2=x.