  We use explicit methods to study the 4-torsion points on the Jacobian variety
of the Fermat quartic. With the aid of computer algebra systems, we explicitly
give a basis of the group of 4-torsion points. We calculate the Galois action,
and show that the image of the mod 4 Galois representation is isomorphic to the
dihedral group of order 8. As applications, we calculate the Mordell-Weil group
of the Jacobian variety of the Fermat quartic over each subfield of the 8-th
cyclotomic field. We determine all of the points on the Fermat quartic defined
over quadratic extensions of the 8-th cyclotomic field. Thus we complete
Faddeev's work in 1960.
