## Abstract

Let *n* be a positive integer. We show that if the equation

has a solution (*x,y,z*) in a cubic number field *K* with \(xyz \neq 0\), then the Galois group of the field *K* is the symmetric group *S*_{3}. In addition, we show that for every positive integer \(d>1\), there exists a number field *K*_{d} of degree *d* such that equation (1) has a solution (*x,y,z*) in *K*_{d} with \(xyz \neq 0\). This paper extends the recent work of Bremner and Choudhry [5].

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## References

A. Aigner, Über die Möglichkeit von \(x^4 + y^4 = z^4\) in quadratische Körper, J. Math. Verein., 43 (1934), 226–228.

W. Bosma, J. Cannon, and C. Playoust, The MAGMA algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235–265.

A. Bremner, Some quartic curves with no points in any cubic field, Proc. London Math. Soc., 52 (1986), 193–214.

A. Bremner, personal communication, Dec. 2021.

A. Bremner and A. Choudhry, The Fermat cubic and quartic curves over cyclic fields, Period. Math. Hungar., 80 (2019), 147–157.

J. W. S. Cassels, The arithmetic of certain quartic curves, Proc. Roy. Soc. Edinburgh Sect. A, 100 (1985) 201–218.

H. Cohen, Number Theory. Vol. I: Tools and Diophantine Equations, Graduate Texts in Mathematics, vol. 239, Springer (2006), pp. 397–410.

D. F. Coray, Algebraic points on cubic hypersurfaces, Acta Arith., 30 (1976), 267–296.

V. A. Demjanenko, The indeterminate equation \(x^6+y^6=az^2, x^6+y^6=az^3, x^4+y^4=az^4\), Amer. Math. Soc. Transl., 199 (1983), 27–34.

L. E. Dickson, History of the Theory of Numbers, Dover Publications (2005), 615–620.

D. K. Faddeev, Group of divisor classes on the curve defined by the equation \(x^4 + y^4 = 1\), Soviet Math. Dokl., 1 (1960), 1149–1151.

G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73 (1983), 349–366.

E. V. Flynn and J. S.Wetherell, Covering collections and a challenge problem of Serre, Acta Arith., 98 (2001), 197–205.

A. Li, The Diophantine equation \(x^4+2^ny^4=1\) in quadratic number fields, Bull. Aust. Math. Soc., 104 (2021), 21–28.

E. D. Manley, On quadratic solutions of \(x^4 + py^4 = z^4\), Rocky Mountain J. Math., 36 (2006), 1027–1031.

L. J. Mordell, Some quartic diophantine equations of genus 3, Proc. Amer. Math. Soc., 17 (1966), 1150–1158.

L. J. Mordell, The Diophantine equation \(x^4+y^4 = 1\) in algebraic number fields, Acta Arith., 14 (1967/1968), 347–355.

V. V. Prasolov, Polynomials, Springer (Berlin, 2010).

J. P. Serre, Lectures on the Mordell–Weil Theorem, Translated and edited by Martin Brown from notes by Michel Waldschmidt, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn (Braunschweig, 1989).

N. X. Tho, Some Diophantine problems, PhD thesis, Arizona State University (2019).

N. X. Tho, Fermat quartics with only trivial solutions in any odd degree number field, Period. Math. Hungar. (2021), https://doi.org/10.1007/s10998-021-00446-2.

## Acknowledgement

The author thanks Professor Andrew Bremner for his careful explanation of the mathematics in Cassels’ paper [6] and for his permission to include the example on the curve \(4x^4 +97y^4 = z^4\) into this paper. The author would like to thank the referee for his careful reading, insightful comments, and valuable suggestions. Part of this work was finished during the author’s stay at the Vietnam Institute of Advanced Study in Mathematics (VIASM). The author would like to thank the Institute for the support.

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The author is supported by the Vietnam National Foundation for Science and Technology Development (grant number 101.04-2019.314).

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Tho, N.X. The equation \(x^4+2^ny^4=z^4\) in algebraic number fields.
*Acta Math. Hungar.* **167**, 309–331 (2022). https://doi.org/10.1007/s10474-022-01226-1

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DOI: https://doi.org/10.1007/s10474-022-01226-1