Friday, January 14, 2011

Thoughts on a QSM proof the Kronecker-Weber Theorem

As I discussed in my last posts, our whole interest in Bost-Connes systems is due to its interaction with the class field theory of certain number fields. In the case of $\mathbb{Q}$, which I'm covering in my MSci project, we can recover the full class field theory and get a description of the maximal abelian/cyclotomic extension of $\mathbb{Q}$ without reference to field extensions. To be precise, we have

Thrm 1 (Prop 3.33 Connes & Marcolli) The quotient $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]/(\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \cap J)$ is a field isomorphic to $\mathbb{Q}^\text{cycl}$ and the ideal $(\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \cap J)$ in $C^\ast(\mathbb{Q}/\mathbb{Z})$ is equal to ideal generated by the $\pi_m$ in $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$, where $J$ and $\pi_m$ are defined in my last post.

Despite this result, Connes & Marcolli state that its still an open problem to find a proof the of the Kronecker-Weber theorem using methods from Quantum Statistical Mechanics. After some googling today, I couldn't find any more information on this, and [perhaps due to my ignorance] I can't figure out why it would be so difficult. I must not understand something. But it would seem to be that given a finite abelian extention of the rationals $\mathbb{K}$ our task would be to find an ideal $K$ such that $(\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \cap J) \subset K$ and so that $\mathbb{K} \sim \mathbb{Q}[\mathbb{Q}/\mathbb{Z}]/K \subset \mathbb{Q}[\mathbb{Q}/\mathbb{Z}]/(\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \cap J) \sim \mathbb{Q}^\text{cycl}$. I obviously don't know much about the details of implementing it, but it might be something to think about as I study the proof of the that main class field theory result of the Bost Connes system. Any thoughts?

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