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A polynomial f in the set {Xⁿ+Yⁿ, Xⁿ +Yⁿ-Zⁿ, Xⁿ +Yⁿ+Zⁿ, Xⁿ +Yⁿ-1} lends itself to an elementary proof of the following theorem: if the coordinate ring over ℚ of f is factorial, then n is one or two. We give a list of problems suggested by this result.
We explore the connection between atomicity in Prüfer domains and their corresponding class groups. We observe that a class group of infinite order is necessary for non-Noetherian almost Dedekind and Prüfer domains of finite character to be atomic. We construct a non-Noetherian almost Dedekind domain and exhibit a generating set for the ideal class semigroup.
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