Let be the graded polynomial algebra over where denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra viewed as a graded left module over the mod- Steenrod algebra, For this problem is still unsolved, even in the case of with the help of computers.
In this paper, we study the hit problem for the case in degree with an arbitrary non-negative integer. By considering as a trivial -module, then the hit problem is equivalent to the problem of finding a basis of -graded vector space The main goal of the current paper is to explicitly determine an admissible monomial basis of the -graded vector space in some degrees. At the same time, the behavior of the sixth Singer algebraic transfer in degree is also discussed at the end of this article. Here, the Singer algebraic transfer is a homomorphism from the homology of the mod- Steenrod algebra, to the subspace of consisting of all the -invariant classes of degree