## Tohoku Mathematical Journal 2009 December SECOND SERIES VOL. 61, NO. 4

 Tohoku Math. J. 61 (2009), 551-570

Title THE IDEAL CLASS GROUP OF THE $\boldsymbol{Z}_p$-EXTENSION OVER THE RATIONALS

Author Kuniaki Horie and Mitsuko Horie

(Received November 4, 2008, revised July 27, 2009)
Abstract. For any prime number $p$, we study local triviality of the ideal class group of the ${\boldsymbol Z}_p$-extension over the rational field. We improve a known general result in such study by modifying the proof of the result, and pursue known effective arguments on the above triviality with the help of a computer. Some explicit consequences of our investigations are then provided in the case $p\leq7$.

2000 Mathematics Subject Classification. Primary 11R29; Secondary 11R18, 11R20, 11R23.

Key words and phrases. Ideal class group, ${\boldsymbol Z}_p$-extension.