Class number problem

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 (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 . 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. Gauss Conjecture (Class number tends to infinity) Gauss Class Number Problem (Low class number lists) For given low class number (such as 1, 2, and 3), Gauss gives lists of imaginary quadratic fields with the given class number and believes them to be complete. Infinitely many real quadratic fields with class number one Gauss conjectures that there are infinitely many real quadratic fields with class number one. 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. Gauss Conjecture Solved, Heilbronn, 1934. Low class number lists Class number 1: solved, Baker (1966), Stark (1967), Heegner (1952). Class number 2: solved, Baker (1971), Stark (1971) Class number 3: solved, Oesterlé (1985) Class numbers h up to 100: solved, Watkins 2004 Infinitely many real quadratic fields with class number one Open.

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