Imaginary quadratic number field: Difference between revisions

Revision as of 02:31, 24 January 2009

This article defines a number field property: a property that can be evaluated for a number field

Definition

An imaginary quadratic number field is a number field obtained as a quadratic extension of the field of rational numbers by the squareroot of a negative square-free number. In other words, it is of the form $Q [\sqrt{D}]$ where $D < 0$ .

Relation with other properties

Weaker properties

Number field with positive algebraic norm

Facts

There are two qualitatively different kinds of imaginary quadratic number fields: those where $D \equiv 1 mod 4$ and where $D \equiv - 1 mod 4$ . When $D \equiv 1 mod 4$ , the ring of integers is:

$Z [\frac{1 + \sqrt{D}}{2}]$

whereas when $D \equiv - 1 mod 4$ , the ring of integers is:

$Z [\sqrt{D}]$ .

Norm-Euclidean imaginary quadratic number fields

Further information: Classification of norm-Euclidean imaginary quadratic number fields, Euclidean equals norm-Euclidean for imaginary quadratic number field

A norm-Euclidean number field is a number field whose ring of integers has the property that the restriction of the algebraic norm to the nonzero elements of this ring give a Euclidean norm on this ring.

The norm-Euclidean imaginary quadratic number fields are those corresponding to values of $D$ in the set:

${- 11, - 7, - 3, - 2, - 1}$ .

It turns out that for an imaginary quadratic number field, the ring of integers is norm-Euclidean if and only if it is Euclidean.

Imaginary quadratic number fields whose ring of integers is a unique factorization domain

The following are the values of $D$ for which the imaginary quadratic number field $Q [\sqrt{D}]$ is a principal ideal domain, or equivalently, is a unique factorization domain:

${- 163, - 67, - 43, - 19, - 11, - 7, - 3, - 2, - 1}$ .

@@ Line 10: / Line 10: @@
 * [[Stronger than::Number field with positive algebraic norm]]
+==Facts==
+There are two qualitatively different kinds of imaginary quadratic number fields: those where <math>D \equiv 1 \mod 4</math> and where <math>D \equiv -1 \mod 4</math>. When <math>D \equiv 1 \mod 4</math>, the ring of integers is:
+<math>\mathbb{Z}\left[\frac{1 + \sqrt{D}}{2}\right]</math>
+whereas when <math>D \ \equiv -1 \mod 4</math>, the ring of integers is:
+<math>\mathbb{Z}[\sqrt{D}]</math>.
+===Norm-Euclidean imaginary quadratic number fields===
+{{further|[[Classification of norm-Euclidean imaginary quadratic number fields]], [[Euclidean equals norm-Euclidean for imaginary quadratic number field]]}}
+A [[norm-Euclidean number field]] is a number field whose [[ring of integers in a number field|ring of integers]] has the property that the restriction of the [[algebraic norm in a number field|algebraic norm]] to the nonzero elements of this ring give a [[Euclidean norm]] on this ring.
+The norm-Euclidean imaginary quadratic number fields are those corresponding to values of <math>D</math> in the set:
+<math>\{ -11, -7, -3, -2, -1 \}</math>.
+It turns out that for an imaginary quadratic number field, the ring of integers is norm-Euclidean if and only if it is Euclidean.
+===Imaginary quadratic number fields whose ring of integers is a unique factorization domain===
+The following are the values of <math>D</math> for which the imaginary quadratic number field <math>\mathbb{Q}[\sqrt{D}]</math> is a [[principal ideal domain]], or equivalently, is a [[unique factorization domain]]:
+<math>\{ -163, -67, -43, -19, -11, -7, -3, -2, -1 \}</math>.