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

Abstract

For an odd prime number p, let h⁻_p denote the relative class number of the pth cyclotomic field . It is conjectured that h⁻_p 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 = 2^{e + 1}ℓ + 1 with e ⩾ 1 and an odd prime number ℓ. For 1 ⩽ e ⩽ 4, we prove that h⁻_p 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, h⁻_p 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.

Keywords:

• computation
• cyclotomic field
• Mersenne prime number
• relative class number

2010 AMS Subject Classification:

• Primary 11R18
• Secondary 11R29

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 [Werl 14].