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)

Research output: Contribution to journalComment/debate

Abstract

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.

Original languageEnglish (US)
Number of pages1
JournalProceedings of the London Mathematical Society
Volume116
Issue number4
DOIs
StatePublished - Apr 1 2018

ASJC Scopus subject areas

  • Mathematics(all)

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