Class number problem for imaginary quadratic fields

In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields ${\displaystyle \mathbb {Q} ({\sqrt {d}})}$ (for negative integers d) having class number n. It is named after Carl Friedrich Gauss. It can also be stated in terms of discriminants. There are related questions for real quadratic fields and for the behavior as ${\displaystyle d\to -\infty }$.

The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed), but effective bounds (and explicit proofs of completeness of lists) are harder.

The problems are posed in Gauss's Disquisitiones Arithmeticae of 1801 (Section V, Articles 303 and 304).

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