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Class number problem
Finding complete list of imaginary quadratic fields having a given class number
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 {\displaystyle n\geq 1} a complete list of imaginary quadratic fields Q ( d ) {\displaystyle \mathbb {Q} ({\sqrt {d}})} (for negative integers d {\displaystyle d} ) having class number n {\displaystyle n} . It is named after Carl Friedrich Gauss.
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It can also be stated in terms of discriminants. There are related questions for real quadratic fields and for the behavior as d → − ∞ {\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). Gauss discusses imaginary quadratic fields in Article 303, stating the first two conjectures, and discusses real quadratic fields in Article 304, stating the third conjecture. The original Gauss class number problem for imaginary quadratic fields is significantly different and easier than the modern statement: he restricted to even discriminants, and allowed non-fundamental discriminants. For imaginary quadratic number fields, the (fundamental) discriminants of class number 1 are: The non-fundamental discriminants of class number 1 are: Thus, the even discriminants of class number 1, fundamental and non-fundamental (Gauss's original question) are: In 1934, Hans Heilbronn proved the Gauss conjecture. Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number.
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