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.
Dedekind, Uber den zussamenhang zwischen der theorie der ideals und de r theorie der hoheren cyclotimy index , Abh
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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.