Proposition :
Every ideal in k[x] (polynomial ring) is a principal ideal
Proof :
Suppose that $I\subseteq K[x]$. Take $p(x)\in I$ such that $p(x)$ is monic and $deg(p(x))$ is minimal over all polinomials of positive degree.
Take any $f(x)\in I$. We can say that $f(x)=q(x)p(x)+r(x)$. Then, $deg(r(x))=0$, that is, $r(x)$ is zero or a constant polynomial.
In the first case, $f(x)\in (p(x))$, so $(p(x))=I$. It is worth mentioning that $(p(x))$ is the set of all multiples of $p(x)$.
In the second case, $r(x)=\alpha \neq 0\in K[x]$. How can I finish this case?