Let $m(x)$ be a monic irreducible polynomial in $Z[x]$. Is there any criterion for the quotient ring $Z[x]/(m(x))$ to be Dedekind domain ?
Thanks.
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6$\begingroup$ It is delicate to characterize when $\mathbf{Z}[x]/(x^n-a)$ is integrally closed if $x^n-a$ irreducible over $\mathbf{Q}$ (e.g., for prime $p$, $\mathbf{Z}[2^{1/p}]$ is integrally closed if and only if ${\rm{ord}}_p(2^p-2) = 1$, for which $p=1093, 3511$ are the known counterexamples), so it is unlikely that there is a useful general sufficient criterion beyond the obvious one of $m$ being an Eisenstein-translate at every prime dividing its discriminant. See Corollary 3.4, Examples 3.5-3.6, Theorem 5.1, and Theorem 5.3 of math.uconn.edu/~kconrad/blurbs/gradnumthy/integersradical.pdf $\endgroup$– nfdc23Commented Mar 26, 2017 at 0:18
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5$\begingroup$ Are you asking for some kind of general reason, or to check this in concrete cases? For example, if this is coming up in a course then a handy sufficient condition is if the discriminant of $m(x)$ is squarefree. In nature that criterion often doesn't apply. A buzzword that you can use to find more information on this is "monogenic ring of integers". For example, googling that term will lead you to the page mathoverflow.net/questions/21267/…. $\endgroup$– KConradCommented Mar 26, 2017 at 0:47
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$\begingroup$ @KConrad: I will look at it. Thanks. $\endgroup$– user106560Commented Mar 26, 2017 at 1:00
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