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It was mentioned to me briefly in passing about a criterion for rings of integers, referred to as Dedekind's Criterion.

The Criterion essentially said that a ring $\mathbb{Z}[\omega]$ (for $\omega=\sqrt[3]{2}$ for example) is $p$-maximal (in it's fraction field I believe) if and only if the greatest common divisor of certain polynomials reduced modulo $p$ was equal to $1$, (or a constant). These polynomial are related to the prime $p$ in question, but I can't recall how.

I know this is terribly vague, but is anyone aware of a result that goes by Dedekind's criterion that sounds like this? I've been searching for it every, but have had no luck locating it or it's formulation. Does anyone have a possible reference?

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I'm a little confused -- have you googled it? Quite a few relevant results come up... – Cam McLeman Jan 25 '12 at 0:03

I think you'll find it on the first page of this link. Also at this link, which is pages 305-306 of Henri Cohen, A Course in Computational Algebraic Number Theory.

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