Timeline for Criterion for Dedekind domains

7 events

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Nov 12, 2017 at 15:21	history	undeleted	user106560
Mar 31, 2017 at 21:08	history	deleted	user106560		via Vote
Mar 26, 2017 at 1:00	comment	added	user106560		@KConrad: I will look at it. Thanks.
Mar 26, 2017 at 0:47	comment	added	KConrad		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/….
Mar 26, 2017 at 0:18	comment	added	nfdc23		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
Mar 25, 2017 at 23:13	review	First posts
Mar 25, 2017 at 23:15
Mar 25, 2017 at 23:12	history	asked	user106560	CC BY-SA 3.0