Cohomology of the Steenrod algebra mod nilpotents

Let A be the mod 2 Steenrod algebra. Its cohomology, H*(A; Z₂) = $Ext*A$(Z₂, Z₂), is the E₂ term of the Adams Spectral Sequence converging to the 2-component of the stable homotopy groups of the spheres. Palmieri [23] gave a complete general description of the cohomology of the Steenrod algebra modulo nilpotent elements using Quillen's result [24]. He described several families of explicit elements.; We prove that the only elements in filtrations 2 and 3 are those in Palmieri's explicit families. We conjecture that there are no elements in filtration 4 other than in Palmieri's explicit families. We have also found a new family in filtration 5, a new family in 6 and two new families in filtration 7 and conjecture that these, together with Palmieri's elements, yield the only elements in filtrations 5, 6, and 7.