Note on the Class Number of the pth Cyclotomic Field, II
Note on the Class Number of the pth Cyclotomic Field, II
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关于第 p 个分圆场的类号的注释，II
DOI：
10.1080/10586458.2016.1230528
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发表时间：
2017
影响因子：
0.5
通讯作者：
H. Ichimura
中科院分区：
文献类型：
--
作者：
S. Fujima;H. Ichimura
对奇素数p，设h−p表示第p阶分圆域的相对类数。证明了当p的形式为p = 2 <$+ 1且素数为奇素数<$时，h−p是奇的，并且已知如果2是模<$的原根，则该猜想成立。在这篇文章中，我们处理一个素数p = 2 e + 1 <$+ 1，其中e <$1和一个奇素数<$。对于1 ∈ ε ε 4，我们借助于计算机证明了当2是模ε的原根时，h−p是奇数。在不假设k的情况下，我们计算并发现，在e = 1，k < 220和p < 237的范围内，h-p只有在k恰好是梅森素数的四种例外情况下才是偶数。此外，计算更大的p与梅森素数，我们发现一个例外。
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 = 2e + 1ℓ + 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.