Symbol length in the brauer group of a field

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 Cm field F. In particular, for a Cm field F, we show that every F central simple algebra of exponent pt is similar to the tensor product of at most len(p^t, F) ≤ 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 pt and degree p^s is similar to the tensor product of at most len(p^t, p^s,K) ≤ len(p^t, L) symbol algebras of degree p^t, where L is a C_m+edL(A)+p^s−t−1 field.