On the symbol length of symbols

Fix a prime p and let F be a field with characteristic not p. Let GF be the absolute Galois group of F, and let μps be the GF -module of roots of unity of order dividing ps in a fixed algebraic closure of F. Let α ∈ Hn(F, μ^⊗n_ps) be a symbol (i.e., α=a1 ∪··· ∪ an where ai ∈ H1(F, μps)) with effective exponent dividing ps−1 (that is p^s−1α=0 ∈ Hn(GF, μ^⊗n_p). In this work, we show how to write α as a sum of symbols coming from Hn(F, μ⊗n ps−1), that is, symbols of the form pγ for γ ∈Hn(F, μ^⊗n_ps). If n>3 and p=2, we assume F is prime to p closed and of characteristic zero. In the case p=2, we also bound the symbol length of a sum of two symbols with effective exponent dividing ps−1.