The unit equation over cyclic number fields of prime degree

Let = 3 be a prime.We show that there are only finitely many cyclic number fields F of degree 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 ‫(ޑ‬ζ 11 ) + .The number field F is called exceptional if it possesses an exceptional unit.Thus λ is exceptional if and only if (λ, 1 -λ) is a solution to the unit equation (1-1), and F is exceptional