TY - JOUR

T1 - Corrigendum to

T2 - 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)

AU - Bremner, Andrew

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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.

AB - 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.

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U2 - 10.1112/plms.12078

DO - 10.1112/plms.12078

M3 - Comment/debate

AN - SCOPUS:85045153713

VL - 116

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 4

ER -