Twisted Class Field Theory

Neukirch has developed explicit and axiomatic class field theory, which applies to both local and global fields. One of the key ingredients in his theory is a $\widehat{\mathbb{Z}}$-extension of the base field, and in the case of ${\mathbb{Q}}_p$, he uses the maximal unramified extension. However ${\mathbb{Q}}_p$ has another $\widehat{\mathbb{Z}}$-extension, which we shall denote by ${\widehat{\mathbb{Q}}}_p$. Thus, it is natural to ask if we could verify all the axioms required by taking ${\widehat{\mathbb{Q}}}_p$ as the central object instead. We prove this is possible and the two reciprocity maps induced from the two distinct $\widehat{\mathbb{Z}}$-extensions are the same.