The hit problem for the Dickson algebra

Let the mod 2 Steenrod algebra, $\mathcal{A}$, and the general linear group, $GL(k,{\mathbb{F} }_2)$, act on $P_{k}:={\mathbb{F} }_2[x_{1},...,x_{k}]$ with $\vert x_{i}\vert=1$ 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 $D_{k}:=(P_{k})^{GL(k, {\mathbb{F} }_2)}$ is $\mathcal{A}$-decomposable in $P_{k}$ 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 $Q_0S^0$ are the elements of Hopf invariant one and those of Kervaire invariant one.