Solubility of the Equation xq + yq = zq over Cyclotomic Fields \Bbb Q(zn) for Some Small: values of q and n

• Autores: Peter Findeisen
• Localización: Monatshefte für mathematik, ISSN 0026-9255, Vol. 145, Nº 3, 2005, págs. 207-227
• Idioma: inglés
• DOI: 10.1007/s00605-005-0309-0
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• Resumen

  • A natural number q = 2 is said to be a Fermat exponent for the nth cyclotomic field , if xyz = 0 is implied by the above equation over . In this paper, the result is obtained that 3 is a Fermat exponent not only for (which is well-known), but also for the wider field , whereas 3 is ¿almost¿ a Fermat exponent for , in the sense that there is (essentially) only one nontrivial solution of Fermat¿s cubic equation which is given by 9th roots of unity. From these results it follows that 12 is a Fermat exponent for , and 9 is a Fermat exponent for . The corresponding statement for n = 8 is also proved, yielding the main result that n is a Fermat exponent for , when 3 ? n ? 14.