Communications of the
Korean Mathematical Society CKMS

Let $k=\Q(\root \of m)$ 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 $p$. \proclaim{Theorem 1} If the discriminant of $k$ has at least three distinct prime divisors, then 2 divides $h$.\endproclaim \proclaim{Theorem 2} If an odd prime $p$ divides $h$, then $p$ divides $B_{1,\chi \omega^{-1}}$, where $\chi$ is the nontrivial character of $k$, and $\omega$ is the Teichm\"uller character for $p$. \endproclaim \proclaim{Theorem 3} Let $h_n$ be the class number of $k_n$, the $n$th layer of the $\Z_p$-extension $k_{\infty}$ of $k$. If $p$ does not divide $B_{1,\chi \omega^{-1}}$, then $p \nmid h_n$ for all $n \geq 0$. \endproclaim

Keywords: class number, Kummer pairing, circular units

MSC numbers: primary 11R11, 11R29, secondary 11R23