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The 3-class groups of \(\mathbb {Q}(\root 3 \of {p})\) and its normal closure

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Abstract

We determine the 3-class groups of \({\mathbb {Q}}(\root 3 \of {p})\) and \(K={\mathbb {Q}}(\root 3 \of {p},\sqrt{-3})\) when \(p\equiv 4,7\bmod 9\) is a prime and 3 is a cube modulo p. This confirms a conjecture made by Barrucand-Cohn, and proves the last remaining case of a conjecture of Lemmermeyer on the 3-class group of K.

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Acknowledgements

The authors would like to thank Franz Lemmermeyer, René Schoof and Yi Ouyang for helpful comments on this article. The authors are partially supported by Anhui Initiative in Quantum Information Technologies (Grant no. AHY150200) and by the Fundamental Research Funds for the Central Universities (Grant No. WK3470000020 and Grant no. WK0010000061 respectively). The second named author is also supported by NSFC (Grant no. 12001510).

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  1. CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui, China

    Jianing Li

  2. School of Mathematics, Hefei University of Technology, Hefei, 230009, Anhui, China

    Shenxing Zhang

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  1. Jianing Li
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Li, J., Zhang, S. The 3-class groups of \(\mathbb {Q}(\root 3 \of {p})\) and its normal closure. Math. Z. 300, 209–215 (2022). https://doi.org/10.1007/s00209-021-02797-5

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  • DOI: https://doi.org/10.1007/s00209-021-02797-5

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