On the symbol length of symbols (Q6551459)

Let \(p\) be a prime and let \(F\) be a field of characteristic not equal to \(p\) and with absolute Galois group \(G_F\). The Bloch-Kato theorem on Milnor \(K\)-theory and Galois cohomology implies that if \(\alpha\) lies in the kernel of the natural map \(H^n(G_F,\mu_{p^r}^{\otimes n})\to H^n(G_F,\mu_{p^{r-t}}^{\otimes n})\) induced by the \(p^t\)-power map on coefficients, then \(\alpha=p^{r-t}(\sum \alpha_i)\) where the \(\alpha_i\) are symbols in \(H^n(G_F,\mu_{p^t}^{\otimes n})\). The minimal number of symbols occurring in such a sum is the \emph{symbol length} of \(\alpha\). The paper considers the case when \(\alpha\) is itself a symbol and gives bounds for the symbol length of \(\alpha\) for various \(n, r, t\) and fields \(F\). In the case when \(p=2\) the author also treats the case where \(\alpha\) is a sum of two symbols.\N\NFor the entire collection see [Zbl 1539.11006].