Field with one element: Difference between revisions

Revision as of 09:27, 13 May 2025

Revision as of 23:15, 16 July 2025

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== Motivations ==

=== Algebraic number theory ===

One motivation for '''F'''₁ comes from [[algebraic number theory]]. [[André Weil|Weil]]'s proof of the [[Riemann hypothesis for curves over finite fields]] starts with a curve ''C'' over a finite field ''k'', which comes equipped with a [[Function field of an algebraic variety|function field]] ''F'', which is a [[field extension]] of ''k''. Each such function field gives rise to a [[Hasse–Weil zeta function]] ''ζ''_''F'', and the Riemann hypothesis for finite fields determines the zeroes of ''ζ''_''F''. Weil's proof then uses various geometric properties of ''C'' to study ''ζ''_''F''.

The field of rational numbers '''Q''' is linked in a similar way to the [[Riemann zeta function]], but '''Q''' is not the function field of a variety. Instead, '''Q''' is the function field of the [[scheme (mathematics)|scheme]] {{nowrap|Spec '''Z'''}}. This is a one-dimensional scheme (also known as an [[algebraic curve]]), and so there should be some "base field" that this curve lies over, of which '''Q''' would be a [[field extension]] (in the same way that ''C'' is a curve over ''k'', and ''F'' is an extension of ''k''). The hope of '''F'''₁{{nbh}}geometry is that a suitable object '''F'''₁ could play the role of this base field, which would allow for a proof of the [[Riemann hypothesis]] by mimicking Weil's proof with '''F'''₁ in place of ''k''.