## 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 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.

Original language | English |
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Pages (from-to) | 413-427 |

Number of pages | 15 |

Journal | Transactions of the American Mathematical Society |

Volume | 368 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2016 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2015 American Mathematical Society.