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
This list is also written:
where D is interpreted as the discriminant (either of the number field or of an elliptic curve with complex multiplication).
This result was first conjectured by Gauss
and essentially proven by Kurt Heegner
. 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.
On the other hand, it is unknown whether there are infinitely many d
> 0 for which Q
) has class number 1. Computational results indicate that there are many such fields.
- Elkies, Noam D. (1999). "The Klein Quartic in Number Theory", in Levy [1999: p.51–101].
- Goldfeld, Dorian. " The Gauss Class Number Problem For Imaginary Quadratic Fields".
- Goldfeld, Dorian (1985). " Gauss' Class Number Problem for Imaginary Quadratic Fields", Bulletin of the American Mathematical Society, 13: 23–37.
- Kenku, M.Q. (1985). “A note on the integral points of a modular curve of level 7”, Mathematika, 32: 45–48.
- Levy, Silvio [editor] (1999). The Eightfold Way: The Beauty of Klein's Quartic Curve (Cambridge University Press).
- Stark, H.M. (1969). " On the gap in the theorem of Heegner", Journal of Number Theory, 1: 16–27.