In an arbitrary-rank valued field, Rν[α] is integrally closed exactly when every repeated residue factor gives a Euclidean remainder of minimal positive Gaussian valuation, or no factor repeats.
Ershov, A Dedeking criterion for arbitrary valuation rings , Doklady Math 74 (2006), 650–652
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A Dedekind's Criterion over Valued Fields
In an arbitrary-rank valued field, Rν[α] is integrally closed exactly when every repeated residue factor gives a Euclidean remainder of minimal positive Gaussian valuation, or no factor repeats.