## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Primary components of the ideal class group of the $\mathbf{Z}_p$-extension over $\mathbf{Q}$ for typical inert primes

Kuniaki Horie

#### Abstract

Let $p$ be an odd prime, $\mathbf Z_p$ the ring of $p$-adic integers, and $l$ a prime number different from $p$. We have shown in [1] that, if $l$ is a sufficiently large primitive root modulo $p^2$, then the $l$-class group of the $\mathbf Z_p$-extension over the rational field is trivial. We shall modify part of the proof of the above result and see, in the case $p\leq 7$, that the result holds without assuming $l$ to be sufficiently large.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 3 (2005), 40-43.

Dates
First available in Project Euclid: 18 May 2005

https://projecteuclid.org/euclid.pja/1116442034

Digital Object Identifier
doi:10.3792/pjaa.81.40

Mathematical Reviews number (MathSciNet)
MR2128929

Zentralblatt MATH identifier
1114.11086

#### Citation

Horie, Kuniaki. Primary components of the ideal class group of the $\mathbf{Z}_p$-extension over $\mathbf{Q}$ for typical inert primes. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 3, 40--43. doi:10.3792/pjaa.81.40. https://projecteuclid.org/euclid.pja/1116442034

#### References

• K. Horie, Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field, J. London Math. Soc. (2) 66 (2002), no. 2, 257–275.