Algebra Seminar talk
Class numbers of quadratic fields
The ideal class group of a Dedekind domain A is the quotient Cl(A) = Id(A) / P(A) of the group of fractional ideals of A, denoted by Id(A), by the subgroup of principal ideals P(A). The class number h(A) is the order of Cl(A), if finite. The class number is an important measure; most notably, if h(A)=1, then A is a principal ideal domain and therefore it is a unique factorization domain.
In this talk we will study briefly the Gauss conjectures for the behaviour of h(K) when K is a quadratic field, and the progress made toward solving the conjectures. We then intend to present some recent results on the class number one problem for the real quadratic fields with discriminant D=(MN)^2+4MN, for odd positive integers M and N.