Class number problem

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of <a href="/facts/Imaginary_quadratic_field/dUnSDN4e">imaginary quadratic fields</a> 
 
 
 
 
 Q
 
 (
 
 
 d
 
 
 )
 
 
 {\displaystyle \mathbb {Q} ({\sqrt {d}})}
 
 (for negative integers d) having <a href="/facts/Class_number_(number_theory)/LLn5Pwj1">class number</a> n. It is named after <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a>. It can also be stated in terms of <a href="/facts/Discriminant_of_an_algebraic_number_field/mvv6MWKj">discriminants</a>. 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.

Class number problem open-in-new

Class number problem