# Polynomial vs Integer Rings

I have a more conceptual question on the differences between polynomial rings and integer rings. I know they are both Euclidean domains, but is one a subset of the other? More precisely are all polynomial rings integer rings or all integer rings polynomial rings?

Herstein's Abstract Algebra suggests that the notion of roots separates integers compared to polynomials, but it didn't seem obvious to me why this is important,

Absolutely no. For instance, the ring $$\mathbb{Z}[X]$$ of polynomials in the variable $$X$$, then $$\mathbb{Z}[X]$$ is indeed an UFD (unique factorization domain, Gauss Lemma) but it's not a PID thus not even an Euclidean Ring. What you actually meant is that $$K[X]$$ is euclidean whenever $$K$$ is a field, now it is more precise. Indeed $$K[X]$$ and $$\mathbb{Z}$$ do enjoy some similiarities but they are quite different: notice that $$K\subsetneqq K[X]$$ and $$\mathbb{Z}$$ does not contain any non trivial field. If you deepen your studies over quotient rings then you will see even more features which holds only on $$K[X]$$