About this scholarly article

Number Theory (Proceedings of the First Conference of the Canadian Number Theory Association held at the Banff Center, Banff, Alberta, April 17–27, 1988) by Mollin, Richard is a scholarly article available to read on EtoBox.

A conjecture of Gauss, the solution of which seems intractable to this day, says there are infinitely many real quadratic fields with class number one. On the other hand, Gauss conjectured that there are only finitely many complex quadratic fields with class number one. The latter conjecture has been affirmatively settled for some time. For an excellent survey of the problem and its solution, the reader is referred to Goldfeld's article [3]. The early part of this century saw the first significant progress toward a solution of Gauss' conjecture for complex quadratic fields. In particular, in 1918, Landau [6] published a result attributed to Hecke. This result showed that Gauss' conjecture for a complex quadratic field follows from the generalized Riemann hypothesis (GRH) for L(S , χ) where χ is a real non-principal character. It is the purpose of this article to give an overview of the solution to the class number one problem for real quadratic fields of Richaud-Degert (R-D) type, under the GRH assumption, and to show that the GRH can now be removed yielding an unconditional result for the more general extended R-D types with the possibility of only one other value having class num

Author

Mollin, Richard

Publisher

De Gruyter

Published

1990

Language

EN