Imaginary quadratic number field

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 $\mathbb {Q} [{\sqrt {D}}]$ 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 for odd $D$: 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:

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

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

$\mathbb {Z} [{\sqrt {D}}]$.

Also of interest is the case where $D\equiv 2\mod 4$.

Norm-Euclidean imaginary quadratic number fields

Further information: Classification of norm-Euclidean imaginary quadratic integer rings, Euclidean equals norm-Euclidean for imaginary quadratic integer ring

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 $\mathbb {Q} [{\sqrt {D}}]$ is a principal ideal domain, or equivalently, is a unique factorization domain:

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