POINTS ON $x^4+y^4=z^4$ OVER QUADRATIC EXTENSIONS OF ${\mathbb {Q}}(\zeta _8)(T_1,T_2,\ldots ,T_n)$

Abstract Ishitsuka et al. [‘Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic’, Int. J. Number Theory 16 (4) (2020), 881–905] found all points on the Fermat quartic ${F_4\colon x^4+y^4=z^4}$ over quadratic extensions of ${\mathbb {Q}}(\zeta _8)$ , where $\zeta _8$ is the eighth primitive root of unity $e^{i\pi /4}$ . Using Mordell’s technique, we give an alternative proof for the result of Ishitsuka et al. and extend it to the rational function field ${\mathbb {Q}}({\zeta _8})(T_1,T_2,\ldots ,T_n)$ .