On the $Z_p$-Extensions Over $Q(\sqrt{m})$
Commun. Korean Math. Soc. 1998 Vol. 13, No. 2, 233-242
Jae Moon Kim
Inha University
Abstract : 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
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