A global structure theorem for the mod 2 Dickson algebras, and unstable cyclic modules over the Steenrod and Kudo-Araki-May algebras

The Dickson algebra Wn+1 of invariants in a polynomial algebra over F-2 is an unstable algebra over the mod 2 Steenrod algebra A, or equivalently, ove r the Kudo-Araki-May algebra K of 'lower' operations. We prove that Wn+1 is a free unstable algebra on a certain cyclic module, module just one additi onal relation. To achieve this, we analyse the inter?,lay of actions over A and K to characterize unstable cyclic modules with trivial action by the s ubalgebra A(n-2) on a fundamental class in degree 2(n) - a. This involves a new family of left ideals J(a) in K, which play the role filled by the ide als A<(A(n-2))over bar> in the Steenrod algebra.