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 $\hat{ \mathbb{ Z}}$-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 $\hat{ \mathbb{ Z}}$-extension, which we shall denote by $\hat{ \mathbb{ Q}}_p$. Thus, it is natural to ask if we could verify all the axioms required by taking $\hat{ \mathbb{ Q}}_p$ as the central object instead. We prove this is possible and the two reciprocity maps induced from the two distinct $\hat{ \mathbb{ Z}}$-extensions are the same.
Categories: math.NT