Is the kernel of $\mathbb Z [x]\to \mathbb R, x\mapsto 1/2+\sqrt{2}$ a principal ideal?
I proved that $(4x^2-4x-7)$ is a subset of the kernel. Now I need to prove that everything in the kernel lies in this ideal. If the polynomial that generates the ideal were monic, I would be able to write $$f(x)=(4x^2-4x-7)q(x)+(ax+b)$$
where $f$ is an arbitrary element of the kernel, and then it would follow that $a$ and $b$ must be zero. But what to do in this case when the leading coefficient isn't 1?
This exercise is given after the chapter on primitive polynomials and stuff like that, but I don't know what exactly to apply and how. I can go to $\mathbb Q[x]$ and do division with remainder there. But then what?