On the unit equation over cyclic number fields of prime degree

Abstract

Let ℓ ≠ 3 be a prime. We show that there are only finitely many cyclic number fields F of degree ℓ for which the unit equationhas solutions. Our result is effective. For example, we deduce that the only cyclic quintic number field for which the unit equation has solutions is Q(ζ 11 ) + .