The Fermat cubic and quartic curves over cyclic fields

Andrew Bremner, Ajai Choudhry

Research output: Contribution to journalArticle

Abstract

First we show that there exist infinitely many distinct cyclic cubic number fields K such that the Fermat cubic x3+ y3= z3 has non-trivial points in K. Second, we show that the Fermat quartic x4+ y4= z4 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 Z/ 4 Z or A4.

Original languageEnglish (US)
JournalPeriodica Mathematica Hungarica
DOIs
StateAccepted/In press - Jan 1 2019
Externally publishedYes

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Keywords

  • Cyclic cubic number field
  • Fermat cubic
  • Fermat quartic

ASJC Scopus subject areas

  • Mathematics(all)

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