## Abstract

Given a field F of char(F) = 2, we define u ^{n} (F) to be the maximal dimension of an anisotropic form in Iq ^{n} F. For n = 1 it recaptures the definition of u(F). We study the relations between this value and the symbol length of H ^{n} _{2} (F), denoted by sl ^{n} _{2} (F). We show for any n ≥ 2 that if 2 ^{n} ≤ u ^{n} (F) ≤ u ^{2} (F) < ∞, then (Formula Presented). As a result, if u(F) is finite, then sl ^{n} _{2} (F) is finite for any n, a fact which was previously proven when char(F) ≠ 2 by Saltman and Krashen. We also show that if sl ^{n} _{2} (F) = 1, then u ^{n} (F) is either 2 ^{n} or 2 ^{n+1} .

Original language | English |
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Pages (from-to) | 513-521 |

Number of pages | 9 |

Journal | Proceedings of the American Mathematical Society |

Volume | 147 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2019 |

## Keywords

- Kato-Milne cohomology
- Quadratic forms
- Symbol length
- U-invariant

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