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I was reading the Wikipedia article about The Field with One Element and I came across the following quotes:

"...$F_1$ 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."

"most proposed theories of $F_1$ replace abstract algebra entirely"

I wonder what would the definitions of Algebraic Structures like fields, vector spaces, groups, rings..etc look like if The Field with One Element does exist?

is "The Field with One Element" itself, if does exist, an Algebraic Structure?

Haider Atrah
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    See also http://mathoverflow.net/questions/2300/what-is-the-field-with-one-element. – lhf Oct 30 '15 at 11:23
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    For your consideration: https://youtu.be/x95hJ6F87fw?t=59m59s – Alex Youcis Oct 30 '15 at 11:29
  • @AlexYoucis - I was thinking of exactly the same thing! – peter a g Oct 30 '15 at 11:47
  • so if this object exists, is it an Algebraic Structure? – Haider Atrah Oct 30 '15 at 12:15
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    @HaiderAtrah I mean, not to be snarky, but what precisely do you even mean by that? What is your definition of an 'algebraic structure'? It does not exist as a scheme in any reasonable sense. It may exist, from what I've heard, in the context of 'generalized rings', but these objects were constructed largely to contain this the theory $\mathbb{F}_1$. The oft cited paper is this one (where the notion was developed). Another good thing to look at if you want to get an idea for the possible uses of $\mathbb{F}_1$ this article of Conne's is nice – Alex Youcis Oct 30 '15 at 12:50
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    click here. Another good reference for ideas is this article. Of course, someone else might be able to say something more reasonable. – Alex Youcis Oct 30 '15 at 12:52
  • @AlexYoucis Forgive my naive question, my background in Abstract Algebra is exteremly shallow. For me Abstract Algebra is the study of Algebraic Structures which generally refers to a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a list of axioms. When I came cross the elusive non-existent " Field with One Element" I struggled to form a mental picture of what this object could look like, if we assume it exists is it an Algberiac Structure in the above sense? it seems I need to cover lots of technicalities before I can understand it – Haider Atrah Oct 30 '15 at 13:36
  • You might want to read this : https://www.amazon.com/Absolute-Arithmetic-F1-geometry-European-Mathematical/dp/3037191570 – Chickenmancer May 06 '17 at 18:21
  • There is an undergrad-level intro to the elusive phantom "field with one element" in week 259 of John Baez's This Week's Finds in Mathematical Physics. – Bill Dubuque Oct 13 '24 at 02:30

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As explained in the linked answers, the notion of a "field with one element" is a catch all term for a system of linked ideas and phenomena throughout algebra, which aren't (yet) described satisfactorily within our current axiomatic system.

So the field with one element is not in any sense a field, or even a set with extra structure, and any proposed definition solely along these lines misses the point, which is to find an algebraic framework that explains the (observed) phenomena of interest. For instance, weakening the field axioms may allow one to build a field with one element, but it doesn't solve the problem of explaining whats actually going on. See this question Why isn't the zero ring the field with one element?

As an example, the Weil conjectures for curves state that for a smooth algebraic curve $C$ over a finite field $\mathbb{F}_q$, the number of points of $C$ defined over $\mathbb{F}_{q^n}$ differs from $q+1$ by at most $2g\sqrt{q^n}$. This is an arithmetic problem, counting the number of solutions to equations defined over finite fields, but it has a beautiful solution using algebraic geometry, utilising the $2$ dimensional geometric object $C\times C$ to do intersection theory.

To my knowledge, one of the main driving forces behind the desire for a theory of a field with one element is to replicate this argument, viewing $\mathbb{Z}$ as an algebra over $\mathbb{F_1}$ (whatever this means), if there was a sufficiently developed theory that worked as expected, so we had an "intersection theory", then an analogous argument could be used to prove the Riemann Hypothesis, which is the analogue of the Weil conjecture for curves.

So a good theory of $\mathbb{F}_1$ would allow one to make sense of $\mathbb{Z}\otimes_{\mathbb{F}_1}\mathbb{Z}$, and would be sufficiently precise to develop intersection theory and the estimates needed to make the above argument work.

This is to say, the motivations are there, but finding the right definitions to encapsulate the properties we are after is absolutely the hard part.

Chris H
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  • The definitions have been made long time ago already. See https://arxiv.org/abs/0909.0069 and also https://arxiv.org/abs/0704.2030. Your tensor product is computed there, and the field with one element sits together with all classical fields in the category of generalized fields. Nothing hypothetical, really. – Martin Brandenburg Oct 13 '24 at 00:43