A note on the relative class number of the cyclotomic Z(p)-extension of Q(root-p)

作者：Ichimura Humio^*; Nakajima Shoichi

来源：Proceedings of the Japan Academy Series A-Mathematical Sciences, 2012, 88(1): 16-20.

DOI：10.3792/pjaa.88.16

• Class number
• quadratic field
• cyclotomic Z(p)-extension
• non-p part

Let p be a prime number with p equivalent to 3 mod 4 and q = (p - 1)/2. Let k = Q(root-P) and k(infinity)/k be the cyclotomic Z(p)-extension. Denote by h(n)(-) the relative class number of the n-th layer k(n). Let l be a prime number with l not equal p. We show that, for any n %26gt;= 1, l does not divide h(n)(-)/h(n-1)(-) (resp. h(1)(-)/h(0)(-)) if l is a primitive root modulo p(2) (resp. p) and l %26gt;= q - 2 (resp. l %26gt;= q - 6). Further, we show with the help of computer that when p %26lt; 10000 and n %26lt;= 100, l does not divide h(n)(-)/h(n-1)(-) (resp. h(1)(-)/h(0)(-)) for any prime l which is a primitive root modulo p(2) (resp. p).

• 出版日期2012-1