### The Field With One Element

#### Posted by David Corfield

For some grand theory building and an answer to the question ‘What is the field with one element?’, see Nikolai Durov’s New Approach to Arakelov Geometry.

There’s something extremely intriguing about a mathematical entity which has known effects, but which has not been defined. It generates a sense of independent reality. As I mentioned in the Tuesday 8 November entry on my old blog, a vector space over the ‘field with one element’ is a pointed set. Thinking in such terms makes sense of many combinatorial facts, see TWF 187.

Here’s Durov’s answer:

The ‘field with one element’ is the free algebraic monad generated by one constant (p. 26) or the universal generalized ring with zero (p. 33).

This will need some unpacking.

Posted at April 17, 2007 12:02 PM UTC
## Re: The Field With One Element

Hmm,