A norm-Euclidean number field is a number field 
whose ring of integers 
is Euclidean with respect to the absolute field
norm. Explicitly, for 
with 
, there is a 
such that
 |
Equivalently, for every 
there is a 
with 
(Cavallar and Lemmermeyer 2000). This condition
implies that 
is a Euclidean ring, but Euclideanity with respect
to another function does not require norm-Euclideanity.
Bagger et al. (2026) determined all 58 complex cubic fields which are norm-Euclidean. Here complex means number
field signature 
. Their negative field
discriminants have absolute value at most 876. Bagger et al. (2026) report
that Claude Opus 5 produced the computational solution, which they audited.
