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Class numbers of pure fields


Author: R. A. Mollin
Journal: Proc. Amer. Math. Soc. 98 (1986), 411-414
MSC: Primary 11R29; Secondary 11R21, 11R32
DOI: https://doi.org/10.1090/S0002-9939-1986-0857932-2
MathSciNet review: 857932
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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}$.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0857932-2
Article copyright: © Copyright 1986 American Mathematical Society

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