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The [n.sup.4]-dimensional Hopf algebra [H.sub.n] (p, q) was introduced in [8], where n [??] 2 is an integer, q [member of] K is a primitive n-th root of unity and p [member of] K.
Since [q.sup.-1] is also a primitive n-th root of unity, one can define another Taft Hopf algebra [A.sub.n] ([q.sup.-1]), which is generated, as an algebra, by [g.sub.1] and [x.sub.1] with relations [g.sup.n.sub.1] = 1, [x.sup.n.sub.1] = 0 and [x.sub.1][g.sub.1] = [q.sup.-1][g.sub.1][x.sub.1].
The first author Chen introduced a Hopf algebra [H.sub.n](p, q) in [8], where p, q [member of] K and q is a primitive n-th root of unity. It was shown there that [H.sub.n](p, q) is isomorphic to a cocycle deformation of the tensor product [A.sub.n](q) [cross product] [A.sub.n]([q.sup.-1]).
where [omega] is the primitive n-th root of unity, e 2[pi]i/n.
If [xi] is a primitive n-th root of unity in some field containing [F.sub.4], then the minimal polynomial of [[xi].sup.s] over [F.sub.4] is