BEHAVIOR OF THE RING CLASS NUMBERS OF A REAL QUADRATIC FIELD

摘要

Let K be a real quadratic field Q(√n) with an integer n - df2 with the field discriminant d of K and f≥1. Q. Mushtaq found an interesting phenomena that any totally negative number ko with ko < 0 and koσ < 0 belonging to the discriminant n, attains an ambiguous number km with kmkm < 0 after a finitely many actions koAj with 0 ≤ j ≤ m by modular transformations Aj ∈ SL2+(Z). Here σ denotes the embedding of K distinct from the identity. In this paper we give a new aspect for the process to reach an ambiguous number from a totally negative or totally positive number, by which the gap of the proof of Q. Mushtaq's Theorem is complemented. Next as an analogue of Gau?' Genus Theory, we prove that the ring class number h+(df2), coincides with the ambiguous class number belonging to the discriminant n - df2 and it's behavior is unbounded, when f with suitable prime factors goes to infinity using the ring class number formula.