# Projective Geometry over $F_1$ and the Gaussian Binomial Coefficients

by Henry Cohn

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?"