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Leonidlednev (talk | contribs) Reverting edit(s) by 205.207.236.253 (talk) to rev. 1239241460 by ShelfSkewed: Vandalism (UV 0.1.5) |
Adding local short description: "Theoretical object in mathematics", overriding Wikidata description "hypothetical entity that behaves like a finite field with a single element" |
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{{Short description|Theoretical object in mathematics}}
{{Use dmy dates|date=September 2020}}
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 '''F'''<sub>1</sub>, or, in a French–English pun, '''F'''<sub>un</sub>.<ref>"[[wikt:un#French|un]]" is French for "one", and [[wikt:fun|fun]] is a playful English word. For examples of this notation, see, e.g. {{harvtxt|Le Bruyn|2009}}, or the links by Le Bruyn, Connes, and Consani.</ref> The name "field with one element" and the notation '''F'''<sub>1</sub> are only suggestive, as there is no field with one element in classical [[abstract algebra]]. Instead, '''F'''<sub>1</sub> refers to the idea that there should be a way to replace [[set (mathematics)|set]]s and [[Operation (mathematics)|operation]]s, the traditional building blocks for abstract algebra, with other, more flexible objects. Many theories of '''F'''<sub>1</sub> have been proposed, but it is not clear which, if any, of them give '''F'''<sub>1</sub> 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 (algebra)|characteristic]] is one.
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