Bremner, Andrew; Choudhry, Ajai The Fermat cubic and quartic curves over cyclic fields. (English) Zbl 1474.14041 Period. Math. Hung. 80, No. 2, 147-157 (2020). Summary: 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 \(A_4\). Cited in 1 ReviewCited in 5 Documents MSC: 14G25 Global ground fields in algebraic geometry 11D25 Cubic and quartic Diophantine equations 11G05 Elliptic curves over global fields Keywords:Fermat cubic; Fermat quartic; cyclic cubic number field PDFBibTeX XMLCite \textit{A. Bremner} and \textit{A. Choudhry}, Period. Math. Hung. 80, No. 2, 147--157 (2020; Zbl 1474.14041) Full Text: DOI References: [1] Abhyankar, SS, Algebraic Geometry for Scientists and Engineers, Mathematical Surveys and Monographs, 35 (1990), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0709.14001 [2] Bremner, A., Some quartic curves with no points in any cubic field, Proc. Lond. Math. Soc., 52, 3, 193-214 (1986) · Zbl 0602.14020 [3] Cassels, JWS, The arithmetic of certain quartic curves, Proc. R. Soc. Edinb. Sec. A, 100, 201-218 (1985) · Zbl 0589.14029 [4] Klassen, M.; Tzermias, P., Algebraic points of low degree on the Fermat quintic, Acta Arith., 82, 393-401 (1997) · Zbl 0917.11022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.