Euclidean Domain - Norm-Euclidean Fields

Norm-Euclidean Fields

Algebraic number fields K come with a canonical norm function on them: the absolute value of the field norm N that takes an algebraic element α to the product of all the conjugates of α. This norm maps the ring of integers of a number field K, say O_K, to the nonnegative rational integers, so it is a candidate to be a Euclidean norm on this ring. If this norm satisfies the axioms of a Euclidean function then the number field K is called norm-Euclidean. Strictly speaking it is the ring of integers that is Euclidean since fields are trivially Euclidean domains, but the terminology is standard.

If a field is not norm-Euclidean then that does not mean the ring of integers is not Euclidean, just that the field norm does not satisfy the axioms of a Euclidean function. Indeed, there are examples of number fields whose ring of integers is Euclidean but not norm-Euclidean, a simple example being the quadratic field . However, for imaginary quadratic fields it is indeed the case that the Eucidean fields are norm-Euclidean. Finding all such fields is a major open problem, particularly in the quadratic case.

The norm-Euclidean quadratic fields have been fully classified, they are where d takes the values

−11, −7, −3, −2, −1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 (sequence A048981 in OEIS).