norm-Euclidean number field
Here N means the norm function in .
Theorem 1. A field is norm-Euclidean if and only if each number of is in the form
where is an of the field and
Proof. First assume the condition (1). Let and be integers of , . Then there are the numbers such that is integer and
Thus we have
Here is integer, since and are integers. We also have
Accordingly, is a norm-Euclidean number field. Secondly assume that is norm-Euclidean. Let be an arbitrary element of the field. We can determine (http://planetmath.org/MultiplesOfAnAlgebraicNumber) a rational integer such that is an algebraic integer of . The assumption guarantees the integers , of such that
Theorem 2. In a norm-Euclidean number field, any two non-zero have a greatest common divisor.
Proof. We recall that the greatest common divisor of two elements of a commutative ring means such a common divisor of the elements that it is divisible by each common divisor of the elements. Let now and be two algebraic integers of a norm-Euclidean number field . According the definition there are the integers and of such that
The ends to the remainder 0, because the numbers form a descending sequence of non-negative rational integers — see the entry norm and trace of algebraic number. As in the Euclid’s algorithm in , one sees that the last divisor is one greatest common divisor of and . N.B. that and may have an infinite amount of their greatest common divisors, depending the amount of the units in .
Remark. The ring of integers of any norm-Euclidean number field is a unique factorization domain and thus all ideals of the ring are principal ideals. But not all algebraic number fields with ring of integers a UFD (http://planetmath.org/UFD) are norm-Euclidean, e.g. .
Theorem 3. The only norm-Euclidean quadratic fields are those with
|Title||norm-Euclidean number field|
|Date of creation||2013-03-22 16:52:26|
|Last modified on||2013-03-22 16:52:26|
|Last modified by||pahio (2872)|