Title of Paper: The hit problem for the Dickson algebra

Author: Nguy\^{e}n H. V. Hung and Tran Ngoc Nam

2000 Mathematics Subject Classification: Primary 55S10, Secondary
55P47, 55Q45, 55T15.


Address of authors: Department of Mathematics, Vietnam National
University, Hanoi 334 Nguyen Trai Street, Hanoi, Vietnam

E-mail address: nhvhung@vnu.edu.vn

E-mail address: namtn@vnu.edu.vn


Abstract: Let the mod 2 Steenrod algebra, ${\cal A}$, and the
general linear group, $GL(k, F_2)$, act on $P_{k}:=
F_2[x_{1},...,x_{k}]$ with $|x_{i}|=1$ in the usual manner. We
prove the conjecture of the first-named author in {\it Spherical
classes and the algebraic transfer}, (Trans. AMS 349 (1997),
3893-3910) stating that every element of positive degree in the
Dickson algebra $D_{k}:=(P_{k})^{GL(k,F_2)}$ is ${\cal
A}$-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.