What is "Field with One Element"? 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?
 A: 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.
