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As far as I know there is many "suggestions" of what should be a "field with one element" $\mathbf{F}_{1}$.

My question is the following:

How we should think or what should be the "absolute Galois group" of the "hypothetical" $\mathbf{F}_{1}$ ?

Community
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Muhammed Ali
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1 Answers1

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The Galois group of the maximal abelian extension of $\mathbb Q$ (or any number field) is given (class field theory) as the quotient of the idele class group by the connected component of the identity which is isomorphic to $\mathbb R$. If there is an ${\mathbb F}_1$ then its extensions provide "constant field extensions" of $\mathbb Q$. So, to be compatible with class field theory and keep the analogy with function fields, $\mathbb R$ must be at least the abelianization of the absolute Galois group of ${\mathbb F}_1$. But finite fields are abelian, so this suggests the answer. A slightly different perspective is that Weil constructed canonically and functorially an extension of the absolute Galois group of any number field by ${\mathbb R}$ (the Weil group) and that should be the extension one would get by allowing constant field extensions again.

Felipe Voloch
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  • $\mathbb F_1$, whatever it is or isn't, is weird. – Wojowu Dec 29 '16 at 12:58
  • I think the “which is isomorphic to $\mathbb{R}$” part is wrong: it should be $\prod_p \mathbb{Z}_p^\times$ (at any rate, it should be profinite and there is torsion). – Gro-Tsen Aug 15 '19 at 12:50
  • @Gro-Tsen I was referring to the connected component of the identity and you seem to be talking about the quotient. – Felipe Voloch Aug 15 '19 at 15:57
  • OK, but then I don't understand “$\mathbb{R}$ must be at least the abelianization of the absolute Galois group of $\mathbb{F}_1$”… – Gro-Tsen Aug 15 '19 at 22:33
  • Imagine that the whole ideal class group somehow describes abelian extensions of $\mathbb{Q}$ in some extended sense which includes "constant field extensions" so we should see the absolute Galois group of $\mathbb{F}_1$ (or its abelianization if it's nonabelian) in the ideal class group. The connected component of the identity is a candidate. – Felipe Voloch Aug 15 '19 at 23:38
  • If we think about a curve over a finite field, then the Galois group of the finite field shows up as the quotient of the total fundamental group, right? So your argument really suggests that the galois group of F_1 should be the units in the profinite completion. In fact the Cyclotomic extensions should be the "constant" extensions as suggested by many things including Iwasawa theory. – Asvin Dec 11 '21 at 17:16
  • @FelipeVoloch could you explain again in a little more détail that sentence "$\mathbb{R}$ must be at least the Abelianization of the absolute Galois group of $\mathbb{F}_{1}$"? – The Thin Whistler Jul 09 '23 at 07:51