A&F point out that some of the therems they have proved for \(\displaystyle \mathbb{Q} [x] \) are false if we restrict ourselves to polynomials with integer coefficients.

For example they point out that the Divsion Theorem is false for \(\displaystyle \mathbb{Z} [x] \).

Also the GCD identity fails in \(\displaystyle \mathbb{Z} [x] \).

They then ask the reader the polynomials 2 and x in \(\displaystyle \mathbb{Z} [x] \) pointing out the 1 is the GCD for these polynomials. A&F then ask the reader to prove that we cannot write 1 as a linear combination of 2 and x

Can anyone help me with a rigorous and formal proof of this.

Peter