Field With One Element

In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F₁, or, in a French–English pun, F_un. The name "field with one element" and the notation F₁ are only suggestive, as there is no field with one element in classical abstract algebra. Instead, F₁ refers to the idea that there should be a way to replace sets and operations, the traditional building blocks for abstract algebra, with other, more flexible objects. While there is still no field with a single element in these theories, there is a field-like object whose characteristic is one.

F₁ cannot be a field because all fields must contain two distinct elements, the additive identity zero and the multiplicative identity one. Even if this restriction is dropped, a ring with one element must be the trivial ring, which does not behave like a finite field. Instead, most proposed theories of F₁ replace abstract algebra entirely. Mathematical objects such as vector spaces and polynomial rings can be carried over into these new theories by mimicking their abstract properties. This allows the development of commutative algebra and algebraic geometry on new foundations. One of the defining features of theories of F₁ is that these new foundations allow more objects than classical abstract algebra, one of which behaves like a field of characteristic one.

The possibility of studying the mathematics of F₁ was originally suggested in 1956 by Jacques Tits, published in (Tits 1957), on the basis of an analogy between symmetries in projective geometry and the combinatorics of simplicial complexes. F₁ has been connected to noncommutative geometry and to a possible proof of the Riemann hypothesis. Many theories of F₁ have been proposed, but it is not clear which, if any, of them give F₁ all the desired properties.