Algebra Seminar talk
2014-05-09
Kostadinka Lapkova
Class numbers of quadratic fields
Abstract:
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.