Class numbers of pure fields

Abstract:

Necessary and sufficient conditions are given for the class number ${h_{{K_i}}}$ of a pure field $K = Q({m^{1/{p^i}}}){\text { }}({\text {for }}i = 1,2)$ to be divisible by ${p^r}$ for a given positive integer $r$ and prime $p$. Moreover the divisibility of ${h_{{K_i}}}$ by $p$ is linked with the $p$-rank of the class group of the $K(\varsigma )$ and prime divisors of $m$, where $\varsigma$ is a primitive $p$th root of unity. Finally we prove in an easy fashion that for a given odd prime $p$ and any natural number $t$ there exist infinitely many non-Galois algebraic number fields (in fact pure fields) of degree ${p^i}(i = 1,2)$ over $Q$ whose class numbers are all divisible by ${p^t}$.