Proof of Relative Class Number One for Almost All Real Quadratic Fields and a Counterexample for the Rest

Let εD = v+u √ D be the fundamental unit of Z[ √ D] with Z being the ordinary integers, or maximal order, in the rational field Q. We prove that for any square-free integer D > 1, with D not dividing u, there exists a prime fD such that the relative class number HD(fD) = hf2 DD/hD = 1, where hD is the ideal class number of Z[ √ D] and hf2 DD is the ideal class number of Z[fD √ D], the order of index fD in the maximal order Z[ √ D] of Q( √ D). For the remaining case we provide a counterexample to class number one. This completely settles an open question left by Dirichet for any real quadratic field. This vastly generalizes recent results in the literature and does so with chiefly results by Thomas Muir from 1874 that have long gone unrecognized.