A recursion formula for the relative class number of the \(p^ n\)-th cyclotomic field (Q1187141)

Let \(p\) be a prime, \(n\) a positive integer, and \(h^ -_ n\) the relative class number of the \(p^ n\)-th cyclotomic field. The author derives the following class number formula: If \(p\geq 3\), \(n\geq 2\) or \(p=2\), \(n\geq 3\), then \(h^ -_ n=p|\text{det} C| h^ -_{n- 1}\), where \(C\) is a special square matrix of order \((p-1)^ 2p^{n- 2}(1/2)\) defined in this article.