Theorem of Stark-Heegner

The theorem of Stark-Heegner is a theorem of the Théorie of the numbers which indicates precisely which imaginary body of numbers quadratic admits a decomposition in factors first single in their ring of entireties. It solves a particular case of the problem of the number of classes of Gauss for the determination of the number of imaginary quadratic bodies which have a number of class fixed given.

That is to say $\ mathbb {Q} \,$ , the whole of the rational numbers, and D a whole without square (i.e., a product of distinct prime numbers) others that 1. Then the algebraic Bodies of numbers $\ mathbb {Q} (\ sqrt {D}) \,$ is a finished extension of $\ mathbb {Q} \,$ , called a quadratic extension. The many classes of $\ mathbb {Q} (\ sqrt {D}) \,$ is the number of classes of equivalence of the Idéaux of $\ mathbb {Q} (\ sqrt {D}) \,$ , where two ideals $\ mathcal {I} \,$ and $\ mathcal {J} \,$ are equivalent if and only if there exist principal ideal ( has ) and ( B ) such as $(a) \ mathcal {I} = (b) \ mathcal {J} \,$ . Thus, $\ mathbb {Q} (\ sqrt {D}) \,$ is a principal ideal Anneau, (and consequently, a single Anneau of factorization) if and only if the number of classes of $\ mathbb {Q} (\ sqrt {D}) \,$ is equal to 1. The theorem of Stark-Heegner can then be establishes as what follows:

If D < 0, then the number of classes of $\ mathbb {Q} (\ sqrt {D}) \,$ is equal to 1 if and only if D = - 1, - 2, - 3, - 7, - 11, - 19, - 43, - 67, or - 163.

This result was conjectured in first by the German Mathématicien Gauss and shown by Kurt Heegner in 1952, although the demonstration of Heegner was not accepted until Harold Stark gives a demonstration in 1967, which Stark showed was actually equivalent to that of Heegner.

If, on another side, D > 0, then one is unaware of if there exists an infinity of body $\ mathbb {Q} (\ sqrt {D}) \,$ with a number of classes equal to 1. The results by calculations indicate that there exists a great number of such bodies.

References

Dorian Goldfeld: The problem of the number of classes for the imaginary quadratic bodies (in English)

Random links: