Imaginary Quadratic Field

An imaginary quadratic field is a quadratic field with a negative integer that is squarefree. Its elements are with . The two embeddings of in the complex plane send to the two complex conjugates and . Neither image is contained in , so is a totally imaginary field. A real quadratic field, by contrast, has two real embeddings.

For , the field norm is

(1)

which is positive for every nonzero . The ring of integers and number field discriminant are given by

(2)

and

(3)

(Conrad).

The following table gives the ring of integers for each of two familiar fields. Here is the squarefree negative integer in .

	ring of integers
	Gaussian integer
	Eisenstein integer

The units of are exactly its elements of field norm 1. These are unless or . The Gaussian integers have four units and the Eisenstein integers have six. In a real quadratic field, the ring of integers instead has infinitely many units (Sutherland 2025).

The class number of equals 1 iff is a unique factorization domain. For example, in ,

(4)

The factors have field norm 4, 9, or 6. Since cannot equal 2 or 3 for , the factors are all irreducible and the two factorizations are distinct even up to units and order. Exactly nine imaginary quadratic fields have class number 1 (Stark 1967). Their radicands are the negatives of the Heegner numbers.