The complete answer to this question has been completely worked out only when K is an imaginary quadratic field or its generalization, a CM-field.

Elliptic units are an analogue for imaginary quadratic fields of cyclotomic units.

The Stark-Heegner theorem identifies the imaginary quadratic fields of class number 1.

However, for imaginary quadratic fields it is indeed the case that the Eucidean fields are norm-Euclidean.

In 1976 Goldfeld provided the key ingredient for the effective solution of Gauss' class number problem for imaginary quadratic fields.

This effective lower bound then allows the determination of all imaginary fields with a given class number after a finite number of computations.

If l is the quadratic extension of k over which G is an inner form, then l is a totally imaginary field.

The numbers given in Table 4.2 are for an imaginary field in, say, January.

Chapter 5 contains Dirichlet's derivation of the class number formula for real and imaginary quadratic fields.

That is, as quotients of the complex plane by some order in the ring of integers in an imaginary quadratic field.