The Fermat cubic and quartic curves over cyclic fields

  • Andrew BremnerEmail author
  • Ajai Choudhry


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 \(A_4\).


Fermat cubic Fermat quartic Cyclic cubic number field 

Mathematics Subject Classification

14G25 11D25 11G05 



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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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesTempeUSA
  2. 2.LucknowIndia

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