On the Class Numbers in the Cyclotomic Z29- and Z31-Extensions of the Field of Rationals

ABSTRACT

Let p be a prime number. For each prime number ℓ, we consider the problem whether ℓ divides class numbers of finite subextensions in the cyclotomic ${\mathbb{Z}}_p$-extension of the field of rationals or not. Even in the case where ℓ is a primitive root modulo p², the problem is not solved in general. In this paper, we focus on the case p = 29 and 31. And we show that ℓ does not divide class numbers of finite subextensions in the cyclotomic ${\mathbb{Z}}_{29}$- and ${\mathbb{Z}}_{31}$-extensions of the field of rationals if a prime number ℓ is a primitive root modulo p².

KEYWORDS:

• class number
• ${\mathbb{Z}}_p$-extension

2010 AMS SUBJECT CLASSIFICATION:

• 11R29
• 11R18

Acknowledgments

The authors would like to thank Professor Hiroki Aoki for suggesting their collaboration. The authors would also like to thank Professor Yoshitaka Hachimori who gave them valuable advice.

Additional information

Funding

The second author was supported by JSPS KAKENHI grant number 16K17580.