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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).

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    In the setting of Mahler's book, ${\mathbf Z}_g$ is the $g$-adic completion of ${\mathbf Z}$ and is isomorphic to ${\mathbf Z}_{p_1} \times \cdots \times {\mathbf Z}_{p_r}$ where $p_1,\dots,p_r$ are the different prime factors of $g$ (their multiplicities don't matter). Thus formally one can just declare ${\mathbf Q}_{g}$ to be the product of the ${\mathbf Q}_{p_i}$'s, likewise with ${\mathbf C}_g$, etc. Whether this leads anywhere nontrivial is a different issue.2011-10-10

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