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