Corrigendum to: Some Quartic Curves with no Points in any Cubic Field (Proceedings of the London Mathematical Society, (1986), s3-52, 2, (193-214), 10.1112/plms/s3-52.2.193)

A computation in Bremner is incorrect. Examples 12, 13 state that the Diophantine equations x⁴ + y⁴ = D,, with D = 4481, 5617, can have no solution in any cubic number field. The presented argument reduced the problem to a system of three simultaneous quartic equations in four variables, which were asserted to have no solution in an appropriate p-adic field (p = 17, 41, respectively). This latter computation is not correct, as can be seen by the following. Example: Let K = Q(θ) be the cubic number field defined by θ³ − θ² + 10θ + 24 = 0. Then (Formula presented.) Example: Let (Formula presented.) be the cubic number field defined by φ³ − 7φ − 256 = 0. Then (Formula presented.) This observation regarding the examples does not affect the remaining results of the paper.