Let the mod 2 Steenrod algebra, .4, and the general linear group, GL(fc,F2), act on Pk := F2[x1
,...,xk] with |x1| = 1 in the usual manner. We prove the conjecture of the first-named author in Spherical
classes and the algebraic transfer, (Trans. Amer. Math Soc. 349 (1997), 3893-3910) stating that every
element of positive degree in the Dickson algebra Dk := (Pk)GL(k'F2) is A-decomposable in Pfc for
arbitrary k > 2. This conjecture was shown to be equivalent to a weak algebraic version of the classical
conjecture on spherical classes, which states that the only spherical classes in Q0S0 are the elements of Hopf
invariant one and those of Kervaire invariant one. ??2001 American Mathematical Society.
