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Concept
Field with one element
Summary
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, Fun. 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. Many theories of F1 have been proposed, but it is not clear which, if any, of them give F1 all the desired properties. While there is still no field with a single element in these theories, there is a field-like object whose characteristic is one.
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 , 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.
In 1957, Jacques Tits introduced the theory of buildings, which relate algebraic groups to abstract simplicial complexes. One of the assumptions is a non-triviality condition: If the building is an n-dimensional abstract simplicial complex, and if k < n, then every k-simplex of the building must be contained in at least three n-simplices. This is analogous to the condition in classical projective geometry that a line must contain at least three points.
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