How do you factorise polynomials in Z[x]? Z is not a field so you can't use the theorem 'a is a root of f <=> (x-a) divides f'
The Attempt at a Solution
Would you map the polynomials from Z[x] to Q[x] by multiplying by 1, since all elements in Z[x] are in Q[x]. Then factorise in Q[x] usiong the theorem above. If the factors have coefficients in Z than map (multiply by 1) these back to Z[x] so you have factorised these polynomials and they exist in Z[x]. If the factors in Q[x] have coefficients not in Z then you can't map back to Z[x] hence these are not factorisable in Z[x].
Is this how you would factorise polynomials in Z[x], formally?