>field with one element >mathematicians investigate it

>>7846555
there is no field with one element, you mong

>>7846656
>assuming fields must have 1!=0
might as well assume that all conjectures are true.

It's not interesting because there are no ring maps out of it and the zero map is the only map into it from any other ring.

>>7846666
It is a field axiom that [math]0 \neq 1 [/math].

>>7846656 #
Mathematicians are investigating the field with one element.

https://en.m.wikipedia.org/wiki/Field_with_one_element

>>7846555
They aren't literally investigating a field with one element, because such a thing doesn't exist.
Instead, they investigate something like extending the category of fields to a larger category which contains an object that [math]behaves[/math] like the field with one element.
>>7846656
That's a bit strong. There is no such object in the category of fields, but that doesn't mean it doesn't exist at all. It's like saying that infinite products of finite groups don't exist because the result isn't finite - you just have to move to the category of all groups and there it is!
>>7846666
If you drop the4 axiom that 0 =/= 1, the object you end up with - the trivial ring - does not behave the way [math]\mathbb{F}_1[/math] should. For example, a vector space over the trivial ring has just one point, but a vector space over the field with one element "should be" a pointed set (projective space over [math]\mathbb{F}_1[/math] is a set). Of course there is nothing wrong with defining [math]\mathbb{F}_1[/math] as you have, it's just that it isn't interesting.

There are a lot of formulae involving a prime p that still make sense in the limit p -> 1. By interpreting these formulae as statements about [math]\mathbb{F}_p[/math], we can recover the statements in the p -> 1 limit as statements about [math]\mathbb{F}_1[/math]. The most powerful consequence of this is the Riemann hypothesis - Weil proved the characteristic p analogue (https://en.m.wikipedia.org/wiki/Local_zeta-function#Riemann_hypothesis_for_curves_over_finite_fields) in 1940, and by copying his proof with finite fields replaced by the field with one element, we should find a proof of the Riemann hypothesis.

There are a lot of ways of constructing on object that behaves like the field with one element in a variety of different situations, as covered in this paper: http://arxiv.org/pdf/0909.0069v1.pdf.

>>7846677
>Jacques Tits
>Jack's tits

Hahahahaha!

>>7848311
In algebraic geometry over a field, k, the base scheme Spec(k) is zero dimensional - it is just a single point. When you do algebraic geometry over the integers, though, we have a problem - Spec(Z) is one dimensional. It has one point for every prime number, plus an extra point for 0 with the strange property that {0} is not a closed subset of Spec(Z). Normally a discrete, countable set of points would be zero dimensional, but algebraic geometry is strange, as demonstrated by the open points like [0] above.
One of the properties [math]\mathbb{F}_1[/math] should have is that Spec(Z) should be defined over [math]Spec(\mathbb{F}_1)[/math] in the same way that, say, a Riemann surface is defined over Spec(C). Hence [math]Spec(\mathbb{F}_1)[/math] is like a more fundamental geometrical object over over which Spec(Z) lives, allowing for techniques from algebraic geometry over fields to be ported over to number theory (see: function field analogy).

One example of this sort of thinking is seen in the theory of spectra from topology (NOT related to Spec above). A spectrum in algebraic topology is a sequence of topological spaces, [math]E_n[/math] with maps [math]\Sigma E_n \rightarrow E_{n+1}[/math], where sigma is the reduced suspension. We can assign a spectrum to a ring, R, called the Eilenburg Mac Lane spectrum,[math]H R[/math], but [math]H \mathbb{Z}[/math] is NOT initial as a spectrum - the sphere spectrum is! This is the spectrum [math]E_n[/math] = [math]\mathbb{S}^n[/math], whose structure maps are the identity.
Hence the sphere spectrum in this situation behaves like [math]\mathbb{F}_1[/math], and if we look at spectra with some sort of (commutative) ring structure - commutative ring spectra - we can do algebraic geometry over this category, called spectral algebraic geometry. This doesn't lead to a proof of the Riemann hypothesis, but it shows the sort of process one could go through to produce [math]\mathbb{F}_1[/math].

>>7848393
Why is a zero dimensional base scheme of such importance?

File: fake_r.jpg (100KB, 467x636px) Image search: [Google]

100KB, 467x636px

>>7848393
Sami-related: What's a good reason a normal person should care about stable homotopy theory?

>>7848358
>>>/global/rules/2

Field with one element is the new irrational number/ imaginary number, it will get willed into existence at some point

>>7848946
I like this.