Stark–Heegner theorem

The Stark–Heegner theorem classifies the imaginary quadratic fields with class number one: their fundamental discriminants are −3, −4, −7, −8, −11, −19, −43, −67, and −163.

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Imaginary quadratic field: A degree-two extension of the rational numbers obtained by adjoining the square root of a negative rational number. These are precisely the number fields classified by the theorem.
Minkowski bound: A theorem bounding the norm of an integral ideal in every ideal class of a number field. It shows that class number one forces every ideal class to contain a small-norm ideal.
Gauss class number problem: The problem of determining imaginary quadratic fields with a specified class number. The class-number-one case is the first finite classification in this broader problem.
Heegner point: A point on a modular elliptic curve constructed from a complex multiplication point on a modular curve. Class-number-one fields provide especially explicit inputs for constructing these points.
Class number two problem: The problem of determining imaginary quadratic fields whose ideal class group has order two. It extends the same classification question beyond the theorem’s class-number-one case.
Class number: The number of ideal classes in the ring of integers of a number field. The theorem selects fields whose ideal classes form a single class.
Dirichlet class number formula: A formula relating a number field’s class number to the residue of its Dedekind zeta function at one. It connects the class number to analytic quantities that can bound possible discriminants.
Carl Friedrich Gauss: A German mathematician whose work founded major parts of modern number theory. Gauss formulated the broader problem of classifying imaginary quadratic forms by class number.
Complex multiplication elliptic curve: An elliptic curve whose endomorphism ring over the complex numbers is larger than the integers. The listed fields give elliptic curves with especially simple complex multiplication.
Real quadratic field: A degree-two extension of the rational numbers obtained by adjoining the square root of a positive nonsquare rational number. The theorem concerns imaginary, not real, quadratic fields; the real case has different behavior.