quadratic imaginary norm-Euclidean number fields
where is an integer of the field. We then can
and therefore . According to the theorem 1 in the parent entry (http://planetmath.org/EuclideanNumberField), and are norm-Euclidean number fields.
, i.e. . The algebraic integers of have now the canonical form with . Let where be an arbitrary element of the field. Choose the rational integer such that is as close to as possible, i.e. with , and the rational integer such that is as close to as possible; then with . Then we can write
The number is an integer of the field, since . We get the estimation
so . Thus the fields in question are norm-Euclidean number fields.
Theorem 2. The only quadratic imaginary norm-Euclidean number fields are the ones in which .
. The integers of are where . We show that there is a number that can not be expressed in the form with an integer of the field and . Assume that where is an integer of the field (). Then and . Because cannot be 0, we have and thus
Therefore can not be a norm-Euclidean number field ( and so on).
. Now . The integers of have the form with . Suppose that . Then and
So also these fields are not norm-Euclidean number fields.
Remark. The rings of integers of the imaginary quadratic fields of the above theorems are thus PID’s. There are, in addition, four other imaginary quadratic fields which are not norm-Euclidean but anyway their rings of integers are PID’s (see lemma for imaginary quadratic fields, class numbers of imaginary quadratic fields, unique factorization and ideals in ring of integers, divisor theory).
|Title||quadratic imaginary norm-Euclidean number fields|
|Date of creation||2013-03-22 16:52:32|
|Last modified on||2013-03-22 16:52:32|
|Last modified by||pahio (2872)|
|Synonym||imaginary quadratic Euclidean number fields|
|Synonym||imaginary Euclidean quadratic fields|