Symbol length in the Brauer group of a field

Abstract:

We bound the symbol length of elements in the Brauer group of a field $K$ containing a $C_m$ field (for example any field containing an algebraically closed field or a finite field), and solve the local exponent-index problem for a $C_m$ field $F$. In particular, for a $C_m$ field $F$, we show that every $F$ central simple algebra of exponent $p^t$ is similar to the tensor product of at most $\operatorname {len}(p^t,F)\leq t(p^{m-1}-1)$ symbol algebras of degree $p^t$. We then use this bound on the symbol length to show that the index of such algebras is bounded by $(p^t)^{(p^{m-1}-1)}$, which in turn gives a bound for any algebra of exponent $n$ via the primary decomposition. Finally for a field $K$ containing a $C_m$ field $F$, we show that every $F$ central simple algebra of exponent $p^t$ and degree $p^s$ is similar to the tensor product of at most $\operatorname {len}(p^t,p^s,K)\leq \operatorname {len}(p^t,L)$ symbol algebras of degree $p^t$, where $L$ is a $C_{m+\operatorname {ed}_L(A)+p^{s-t}-1}$ field.