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# Stark–Heegner theorem

In number theory, a branch of mathematics, the Stark–Heegner 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 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 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 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

$d = -1, -2, -3, -7, -11, -19, -43, -67, -163.,$

This list is also written:

$D = -3, -4, -7, -8, -11, -19, -43, -67, -163,,$

where D is interpreted as the discriminant (either of the number field or of an elliptic curve with complex multiplication).

## History

This result was first conjectured by Gauss and essentially proven by Kurt Heegner in 1952. Heegner's proof had some minor gaps and was not accepted until Harold Stark gave a complete proof in 1967, which Stark showed was actually equivalent to Heegner's. Heegner died unrecognized. Stark formally filled in the gap in Heegner's proof in 1969. Alan Baker gave a completely different proof at about the same time (or more precisely reduced the result to a finite amount of computation).

In 1985, Monsur Kenku gave a novel proof using the Klein quartic. Noam Elkies gives an exposition of this result.

## Real case

On the other hand, it is unknown whether there are infinitely many d > 0 for which Q(√d) has class number 1. Computational results indicate that there are many such fields.

## References

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