Stark–Heegner theorem

In number theory, the Heegner theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number.

Let Q denote the set of rational numbers, and let d be a square-free integer. The field Q(√d) is a quadratic extension of Q. The class number of Q(√d) is one if and only if the ring of integers of Q(√d) is a principal ideal domain. The Baker–Heegner–Stark theorem can then be stated as follows:

If d < 0, then the class number of Q(√d) is one if and only if ${\displaystyle d\in \{\,-1,-2,-3,-7,-11,-19,-43,-67,-163\,\}.}$

These are known as the Heegner numbers.

By replacing d with the discriminant D of Q(√d) this list is often written as:

${\displaystyle D\in \{-3,-4,-7,-8,-11,-19,-43,-67,-163\}.}$