TITLE: What is the field with one element?
QUESTION [100 upvotes]: I've heard of this many times, but I don't know anything about it.
What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is one-dimensional, namely $\mathop{\text{Spec}}\mathbb Z$.
So, what is the field with one element? And, what are typical geometric objects that descend to $\mathbb F_1$?

REPLY [10 votes]: One of the many resources pointed out above linked to the unpublished preprint by Kapranov and Smirnov called Cohomology determinants and reciprocity laws: the number field case. It's posted page by page in jpgs, but it is definitely worth looking at. They work out the details of vector spaces over F_1^n in detail and also relate the classical power residue symbol to determinants of morphisms of these vector spaces.