Why are $p$-adic integers integers

Question

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?

Answer 1

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.

Answer 2

Alternatively, the $p$-adic integers $\mathbb{Z}_p$ can be defined as the projective limit of the sequence $$ \cdots\longrightarrow\frac{\mathbb{Z}}{p^3\mathbb{Z}} \longrightarrow\frac{\mathbb{Z}}{p^2\mathbb{Z}} \longrightarrow\frac{\mathbb{Z}}{p\mathbb{Z}}, $$ i.e. the ring of sequences $\{a_n\}_{n\geq1}$ where $a_n\in\frac{\mathbb{Z}}{p^{n}\mathbb{Z}}$ and $a_{n+1}\equiv a_n\bmod p^n$.

They are the integers in $\mathbb{Q}_p$ in the sense that they are the closure of $\mathbb{Z}$ in the $p$-adic topology.

In fact, this algebraic definition is independent of that of $\mathbb{Q}_p$ in the sense that you could first define $\mathbb{Z}_p$ as above, prove that is complete and contains $\mathbb{Z}$ as a dense subring, and then obtain $\mathbb{Q}_p$ as its field of fractions.