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)

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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

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Cubic Fields
Quartic
Number field
Curve
P-adic Fields
Diophantine equation

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

@article{a2645d5d9a3346e7a1e64fed91d4e26c,
title = "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)",
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.",
author = "Andrew Bremner",
year = "2018",
month = "4",
day = "1",
doi = "10.1112/plms.12078",
language = "English (US)",
volume = "116",
journal = "Proceedings of the London Mathematical Society",
issn = "0024-6115",
publisher = "Oxford University Press",
number = "4",

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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 -