Before Theorem 6 (in the published version) we claim that C_17 cannot appear as a torsion group over a cyclic and that this follows from M. Derickx, S. Kamienny and B. Mazur, "Rational families of 17-torsion points of elliptic curves over number fields", however this is not true. There are examples of elliptic curves over cyclic quartic fields with C_17 torsion found by Mark van Hoeij and Daeyeol Jeon inedpendently. So the non-existence of C_17 as a torsion group over Q(\zeta_5) should be considered unproven (though it is most likely true). This mistake has been pointed out to me by Maarten Derickx and Andreas Schweizer independently.