Solubility of the Equation x^q + y^q = z^q over Cyclotomic Fields ${\Bbb Q}(\zeta_n)$ for Some Small Values of q and n

Abstract.

A natural number q ≥ 2 is said to be a Fermat exponent for the nth cyclotomic field ${\Bbb Q}(\zeta_n)$, if xyz = 0 is implied by the above equation over ${\Bbb Q}(\zeta_n)$. In this paper, the result is obtained that 3 is a Fermat exponent not only for ${\Bbb Q}(\zeta_3)$ (which is well-known), but also for the wider field ${\Bbb Q}(\zeta_12)$, whereas 3 is “almost” a Fermat exponent for ${\Bbb Q}(\zeta_9)$, 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 ${\Bbb Q}(\zeta_12)$, and 9 is a Fermat exponent for ${\Bbb Q}(\zeta_9)$. The corresponding statement for n = 8 is also proved, yielding the main result that n is a Fermat exponent for ${\Bbb Q}(\zeta_n)$, when 3  n  14.