143
Views
3
CrossRef citations to date
0
Altmetric
Original Articles

Note on the Class Number of the pth Cyclotomic Field, II

ORCID Icon & ORCID Icon
 

Abstract

For an odd prime number p, let hp denote the relative class number of the pth cyclotomic field . It is conjectured that hp is odd when p is of the form p = 2ℓ + 1 with an odd prime number ℓ, and it is known that the conjecture is valid if 2 is a primitive root modulo ℓ. In this article, we handle a prime number p of the form p = 2e + 1ℓ + 1 with e ⩾ 1 and an odd prime number ℓ. For 1 ⩽ e ⩽ 4, we prove that hp is odd whenever 2 is a primitive root modulo ℓ with the help of computer. Without the assumption on ℓ, we compute and find that, in the range e ⩾ 1, ℓ < 220 and p < 237, hp is even only for four exceptional cases, where ℓ happened to be a Mersenne prime number. Further, computing for larger p with a Mersenne prime ℓ, we find one more exception.

2010 AMS Subject Classification:

Acknowledgments

We are grateful to the referee for turning our attention to Washington’s group of cyclotomic units and for informing us of the article [CitationWerl 14].

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.