Euclidean Window

The Euclidean window of a nonzero prime ideal in the ring of integers of a number field is the set of real weights for which the multiplicative weighted norm, for ,

(1)

is a Euclidean function, with (Cardon 2026). Here is the field norm, is the ideal norm, and is the exponent of in the principal ideal . A Euclidean function permits division with strictly smaller remainder and has only finitely many distinct values in each bounded interval.

For and , the ideal norm is 2. Cardon (2026) proved that the Euclidean window is , where

(2)

Thus this nonempty window is not open. Cardon (2026) also obtained the integer-valued, completely multiplicative Euclidean function

(3)

for , with . The usual field norm is not itself a Euclidean function on .

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