Complete classification of the torsion structures of rational elliptic curves over quintic number fields

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⊆H could appear such that H=E(K)tors, for [K:Q]=5. In particular, we prove that at most there is one quintic number field K such that the torsion grows in the extension K/Q, i.e., E(Q)tors⊊E(K)tors.