Consider the approach proposed in the mathoverflow discussion [Elementary Aspects of Galois Deformation](https://mathoverflow.net/q/16775/35619) and write a complete proof if possible. - [ ] Prove Kronecker-Weber following this approach. Useful references: - Section 8.2 of [Modularity lifting beyond the Taylor–Wiles method](https://doi.org/10.1007/s00222-017-0749-x), [Correction](https://doi.org/10.1007/s00222-021-01095-5) - Part 2 of [Motives and L-functions](https://dx.doi.org/10.4310/CDM.2018.v2018.n1.a2) - [There are no unramified abelian extensions of Q (almost)](https://www.galoisrepresentations.com/2012/11/27/there-are-no-unramified-abelian-extensions-of-q-almost/)
Consider the approach proposed in the mathoverflow discussion Elementary Aspects of Galois Deformation and write a complete proof if possible.
Useful references: