When constructing the $p$-adic numbers, we proceed for instance as when constructing $\mathbb{R}$ for the usual distance. Then the integers are king of “natural", we are used to them (are we can see them as the rational algebraic numbers, but this also relies on the specific choice of $\mathbb{Z}$ which I cannot motivate, except maybe as the additive group generated by $1$?)
In the case of $p$-adic numbers, the same process with the $p$-adic topology leads to $\mathbb{Q}_p$. However, it is often defined right after that $\mathbb{Z}_p$ is the ring of $p$-adic integers, e.g. as the unit ball or equivalently as the Laurent series that are power series in $p$.
Is there a deeper definition of $\mathbb{Z}_p$? Can we see them as the algebraic integers of $\mathbb{Q}_p$ for instance? Why are they termed `ìntegers", beyond the analogy of the positive powers appearing only in archimedean integers?
Best Answer
In complex analysis we can speak about meromorphic functions on the plane being holomorphic at a specific point (say, at $0$). But we can also speak about the functions meromorphic or holomorphic near a chosen point without those functions being assumed to come from a meromorphic function on the whole complex plane.
That is the kind of analogy Hensel had in mind: meromorphic functions on the plane (or Riemann sphere) are analogous to rational numbers, $p$-adic numbers are analogous to meromorphic functions at a point (no assumption they make sense "elsewhere"), and $p$-adic integers are analogous to holomorphic functions at a point (no assumptions that they make sense "elsewhere").
By using the label "integer" or "integral" for the valuation ring at a prime (whether it's for the $p$-adic valuation on $\mathbf Q_p$ or the $p$-adic valuation on $\mathbf Q$) we can then say that a rational number is an integer (a good old-fashioned ordinary integer: an element of $\mathbf Z$) iff it is $p$-adically integral for each prime $p$. So this sounds like a local-global idea. And this really is how some rational numbers can be proved to be integers: show they are $p$-adically integral for each $p$. If I want to do this by taking limits in my arguments then I might prefer to be in a complete space when doing this, since limits are better behaved in complete spaces, e.g., I might express my rational number as a $p$-adic integral.