Abstract
First we show that there exist infinitely many distinct cyclic cubic number fields K such that the Fermat cubic \(x^3 + y^3 = z^3\) has non-trivial points in K. Second, we show that the Fermat quartic \(x^4 + y^4 = z^4\) can have no non-trivial points in any cyclic cubic number field. It remains an open question whether the Fermat quartic has any points in quartic number fields with Galois group of type \(\mathbb {Z}/4\mathbb {Z}\) or \(A_4\).
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Acknowledgements
The authors wish to thank the Harish-Chandra Research Institute, Allahabad, for the warm hospitality while this paper was being prepared. The second author also thanks the Harish-Chandra Research Institute for the facilities provided to him to pursue research work in mathematics.
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Bremner, A., Choudhry, A. The Fermat cubic and quartic curves over cyclic fields. Period Math Hung 80, 147–157 (2020). https://doi.org/10.1007/s10998-019-00297-y
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DOI: https://doi.org/10.1007/s10998-019-00297-y