Even if the number field is finite, there are an infinite number of polynomials for it. For example the finite field F2 has two elements {0, 1} but an infinite number of polynomials 0, 1, x, x + 1, x^2, x^2 + x + 1, x^2 + 1, «x^2 + x, and on and on. Some of these have nontrivial factors (x^2 + x = x(x+1)) and some don't (x^2 + x + 1 has no factors beyond itself and 1).
(The other answer to your question is more complete but also a bit more advanced, figured this was worth surfacing.)
(The other answer to your question is more complete but also a bit more advanced, figured this was worth surfacing.)