- On the $Z_p$-extensions over $Q(sqrt{m})$
- Kim, Jae-Moon ;
- Communications of the Korean Mathematical Society, volume 13, issue 2, 1998, Pages 233~242
Abstract
Let $k = Q(\sqrt{m})$ be a real quadratic field. In this paper, the following theorems on p-divisibility of the class number h of k are studied for each prime pp. Theorem 1. If the discriminant of k has at least three distinct prime divisors, then 2 divides h. Theorem 2. If an odd prime p divides h, then p divides $B_{a,\chi\omega^{-1}}$, where $\chi$ is the nontrivial character of k, and $\omega$ is the Teichmuller character for pp. Theorem 3. Let $h_n$ be the class number of $k_n$, the nth layer of the $Z_p$-extension $k_\infty$ of k. If p does not divide $B_{a,\chi\omega^{-1}}$, then $p \notmid h_n$ for all $n \geq 0$.