Stark–Heegner theorem

In number theory, a branch of mathematics, the Baker-Heegner-Stark theorem states precisely which quadratic imaginary number fields admit unique factorisation in their ring of integers. 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 non-square integer. Then Q(√d) is a finite extension of Q of degree 2, called a quadratic extension. The class number of Q(√d) is the number of equivalence classes of ideals of the ring of integers of Q(√d), where two ideals I and J are equivalent if and only if there exist principal ideals (a) and (b) such that (a)I = (b)J. Thus, the ring of integers of Q(√d) is a principal ideal domain (and hence a unique factorization domain) if and only if the class number of Q(√d) is equal to 1. The Baker-Heegner-Stark theorem can then be stated as follows:

These are known as the Heegner numbers.

This list is also written, replacing −1 with −4 and −2 with −8 (which does not change the field), as:

where D is interpreted as the discriminant (either of the number field or of an elliptic curve with complex multiplication). This is more standard, as the D are then fundamental discriminants.