MATH Seminar
| Title: Symbol length over $C_r$ fields |
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| Seminar: Algebra and Number Theory |
| Speaker: Eli Matzri of University of Virginia |
| Contact: Skip Garibaldi, skip@mathcs.emory.edu |
| Date: 2013-05-01 at 3:00PM |
| Venue: W306 |
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| Abstract: A field $F$ is called $C_r$ if every homogenous form of degree $n$ in more then $n^r$ variables has a non-trivial solution. Consider a central simple algebra $A$ of exponent $n$ over a field $F$. By the Merkurjev-Suslin theorem assuming $F$ contains a primitive $n$-th root of unity, $A$ is similar to the product of symbol algebras. The smallest number of symbols required is called the \emph{length} of $A$ and is denoted $l(A)$. If $F$ is $C_r$ we prove $l(A) \leq n^{r-1}-1$. In particular the length is independent of the index of $A$. |
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