Stark-Heegner theorem

The Stark-Heegner theorem is a theorem in number theory that tells 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 square-free integer (i.e., a product of distinct primes) other than 1. Then Q(√ d) is a finite extension of Q, called a quadratic extension. The class number of Q(√ d) is the number of equivalence classes of idealss 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, 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 Stark-Heegner theorem can then be stated as follows:

If d < 0, then the class number of Q(√ d) is equal to 1 if and only if d = −1, −2, −3, −7, −11, −19, −43, −67, or −163.

This result was first conjectured by Gauss and proven by Kurt Heegner in 1952, although Heegner's proof was not accepted until Harold Stark gave a proof in 1967, which Stark showed was actually equivalent to Heegner's.

If, on the other hand, d > 0, then it is unknown whether there are infinitely many fields Q(√ d) with class number 1. Computational results indicate that there are a great many such fields to say the least.