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 S3. In addition, we show that for every positive integer \(d>1\), there exists a number field Kd of degree d such that equation (1) has a solution (x,y,z) in Kd with \(xyz \neq 0\). This paper extends the recent work of Bremner and Choudhry .
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The author thanks Professor Andrew Bremner for his careful explanation of the mathematics in Cassels’ paper  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.
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