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| Format: | Preprint |
| Published: |
2015
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1506.05365 |
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Table of Contents:
- 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.