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 K containing a C m 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 C_m field F F . In particular, for a C m C_m field F F , we show that every F F central simple algebra of exponent p t p^t is similar to the tensor product of at most len ⁡ ( p t , F ) ≤ t ( p m − 1 − 1 ) \operatorname {len}(p^t,F)\leq t(p^{m-1}-1) symbol algebras of degree p t 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 ) (p^t)^{(p^{m-1}-1)} , which in turn gives a bound for any algebra of exponent n n via the primary decomposition. Finally for a field K K containing a C m C_m field F F , we show that every F F central simple algebra of exponent p t p^t and degree p s p^s is similar to the tensor product of at most len ⁡ ( p t , p s , K ) ≤ len ⁡ ( p t , L ) \operatorname {len}(p^t,p^s,K)\leq \operatorname {len}(p^t,L) symbol algebras of degree p t p^t , where L L is a C m + ed L ⁡ ( A ) + p s − t − 1 C_{m+\operatorname {ed}_L(A)+p^{s-t}-1} field.