Imaginary quadratic 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 $$D < 0$$.

Weaker properties

 * Stronger than::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
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:

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