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Projective Geometry over \(F_1\) and the Gaussian Binomial Coefficients

Year of Award: 2005

Publication Information: The American Mathematical Monthly, vol. 111, no. 6, June/July 2004, pp. 487-495.

Summary: There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics and algebra, but it is rarely discussed as such. The purpose of this article is to bring it to the attention of a broader audience, as the solution to a puzzle arising from the question, "What form does the binomial theorem take in a noncommutative ring?"

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About the Author [from The American Mathematical Monthly, v. 111, no. 6, (2004)]

Henry Cohn is a researcher in the theory group at Microsoft Research. He received his Ph.D. from Harvard University in 2000 under Noam Elkies, after which he spent a year as a postdoc at Microsoft Research before joining the group long term in 2001. His primary mathematical interests are number theory, combinatorics, and the theory of computation.

 

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Henry Cohn
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Henry Cohn
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Publication Date: 
Tuesday, September 23, 2008
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There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics and algebra, but it is rarely discussed as such. The purpose of this article is to bring it to the attention of a broader audience, as the solution to a puzzle arising from the question, "What form does the binomial theorem take in a noncommutative ring?"

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