  We classify the possible torsion structures of rational elliptic curves over
quintic number fields. In addition, let E be an elliptic curve defined over Q
and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given
G, which possible groups G \subseteq H could appear such that H=E(K)_tors, for
[K:Q]=5. In particular, we prove that at most there is a quintic number field K
such that E(Q)_tors\neq E(K)_tors.
