### THE HIT PROBLEM FOR THE POLYNOMIAL ALGEBRA OF FIVE VARIABLES IN DEGREE SEVENTEEN AND ITS APPLICATION

#### Abstract

Let Pk := F2[x1, x2,...,xk] be the polynomial algebra in k variables with the degree of each xi being 1, regarded as a module over the mod-2 Steenrod algebra A, and let GLk be the general linear group over the prime field F2. We study the Peterson hit problem of finding a minimal set of generators for the polynomial algebra Pk as a module over the mod-2 Steenrod algebra, A. The results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-2 Steenrod algebra, TorA k,k+n(F2, F2), to the subspace of F2 ⊗A Pk consisting of all the GLk-invariant classes of degree n. In this paper, we explicitly determined the Peterson hit problem for k = 5 and the dgree 17. Using this result, we show that, Singer’s conjecture for the fifth algebraic transfer is true in this degree.

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