Analogous to the $p$-adic ring $\mathbb{Z}_p$, you can (at least formally), define the $g$-adic ring $\mathbb{Z}_g$, where $g$ is composite. Of course when completing to a field, you get in trouble because of the zero-divisors:
My questions are:
- Can something be done here to resolve this (i.e. weaken field axioms, take into account the zero divisors, etc).
- Can you speak of objects like $\mathbb{Q}_g$ and $\mathbb{C}_g$, where $g$ is composite ?
- Can you speak of $\mathbb{A}_{Berk,\mathbb{C}_g}^1$ and $\mathbb{P}_{Berk,\mathbb{C}_g}^1$, where $g$ is composite ?
- Does someone know of research to these objects ?
As vague as ever, I conclude ...
Edit: I found a book "p-adic numbers and their functions" by Kurt Mahler which says something of the field $\mathbb{Q}_p$ and the ring $\mathbb{Q}_g$. Of course you can't algebraically complete a ring like we normally would, but then the question remains: can you construct something which looks like it (an "algebraic completion" of the ring $\mathbb{Q}_g$, so to speak).