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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?

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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,

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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 $K[X]$

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  • $\begingroup$ Thanks for the answer. Would you say that quotient rings are important to study to understand varieties in algebraic geometry? $\endgroup$
    – Evan Kim
    Jan 26, 2021 at 19:00
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    $\begingroup$ Ring theory by itself uses a lot the idea of quotient ring. Unfortunately I never studied any topic in Algebraic Geometry, but I now it involves many ideas over rings (local rings for instance), so my guess is yes you should know something about it. $\endgroup$ Jan 26, 2021 at 19:03

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