The Fermat cubic and quartic curves over cyclic fields

Andrew Bremner; Ajai Choudhry

doi:10.1007/s10998-019-00297-y

The Fermat cubic and quartic curves over cyclic fields

Andrew Bremner, Ajai Choudhry

Mathematical and Statistical Sciences, School of (SoMSS)

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

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 language	English (US)
Pages (from-to)	147-157
Number of pages	11
Journal	Periodica Mathematica Hungarica
Volume	80
Issue number	2
DOIs	https://doi.org/10.1007/s10998-019-00297-y
State	Published - Jun 1 2020

Keywords

Cyclic cubic number field
Fermat cubic
Fermat quartic

ASJC Scopus subject areas

Mathematics(all)

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10.1007/s10998-019-00297-y

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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.",

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AB - 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.

KW - Cyclic cubic number field

KW - Fermat cubic

KW - Fermat quartic

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The Fermat cubic and quartic curves over cyclic fields

Abstract

Keywords

ASJC Scopus subject areas

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