Prinicpal ideal generated by a monic polynomial

Question

Let $R=k$ be a field. Prove that every nonzero (prinicpal) ideal in $k[x]$ is generated by a unique monic polynomial.

I'm struggling to prove this result. Any help or suggestions is much appreciated.

Answer 1

Hint: since nonzero multiples of a polynomial$~P$ necessarily have a degree at least $\deg(P)$, the monic polynomial generating your ideal$~I$ must have the lowest possible degree among elements of $I\setminus\{0\}$.

Show there is at least one monic polynomial among those lowest degree elements; then pick one, and call it $P$. Now for any $Q\in I$ you must show $P$ divides $Q$. To show this, do Euclidean division of $P$ by $Q$, and show that the remainder $R$ is necessarily zero. (This is where the choice of $P$ comes into play.)

Finally, it is clear that the multiples of $P$ with the same degree as$~P$ are just its scalar multiples. And among those, $P$ itself it the only monic one. This shows the uniqueness of the choice of$~P$.