## Note on the class number of the $$p$$th cyclotomic field. III.(English)Zbl 1427.11114

Summary: Let $$p=2\ell^f+1$$ be a prime number with $$f \geqslant 2$$ and an odd prime number $$\ell$$. For $$0 \leqslant t \leqslant f$$, let $$K_t$$ be the imaginary subfield of the $$p$$th cyclotomic field $$\mathbb{Q}(\zeta_p)$$ with $$[K_t : \mathbb{Q}]=2\ell^t$$. Denote by $$h_{p,t}^-$$ the relative class number of $$K_t$$, and by $$h_{p,t}^+$$ the class number of the maximal real subfield $$K_t^+$$. It is known that the ratio $$h_{p,f}^-/h_{p,f-1}^-$$ is odd (and hence so is $$h_{p,f}^+/h_{p,f-1}^+$$) whenever 2 is a primitive root modulo $$\ell^2$$. We show that $$h_{p,f}^+/h_{p,f-1}^+$$ is odd under a somewhat milder assumption on $$\ell$$ and that the ratio $$h_{p,f-1}^-/h_{p,f-2}^-$$ is always odd when $$\ell=3$$.
For Part II, see [S. Fujima and the author, Exp. Math. 27, No. 1, 111–118 (2018; Zbl 1427.11113)].

### MSC:

 11R18 Cyclotomic extensions 11R29 Class numbers, class groups, discriminants

### Keywords:

relative class number; cyclotomic field

Zbl 1427.11113
Full Text:

### References:

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