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

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 $p\leq7$.