Let E be an elliptic curve defined over Q. The torsion subgroup of E over the compositum of all quadratic extensions of Q was studied by Michael Laska, Martin Lorenz, and Yasutsugu Fujita. Laska and Lorenz described a list of 31 possible groups and Fujita proved that the list of 20 different groups is complete.

In this talk, we will generalize the results of Laska, Lorenz and Fujita to the elliptic curves defined over a quadratic cyclotomic field i.e. Q(i) and Q(\sqrt{-3}).