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.[1] 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 does, 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.
^"un" is French for "one", and fun is a playful English word. For examples of this notation, see, e.g. Le Bruyn (2009), or the links by Le Bruyn, Connes, and Consani.
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primitive element theorem, which does not hold true for fields of non-zero characteristic. If a simple extension K(s) / K is not finite, the field K(s) is...
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