[Math] Connected Components of p-adic rationals

general-topologynumber theoryp-adic-number-theory

Notation:

  • $p$ – a prime integer,
  • $\Bbb{Z}_p$ – set of $p$-adic integers,
  • $\Bbb{Q}_p$ – set of $p$-adic rationals,
  • $\Bbb{Q}$ – set of rationals,
  • $\Bbb{R}$ – set of reals.

While reading up on $p$-adic numbers I came to know that $\Bbb{Z}_p$ is both open and closed in $\Bbb{Q}_p$. Since $\Bbb{Z}_p$ is properly contained in $\Bbb{Q}_p$ and is also a clopen set in $\Bbb{Q}_p$ we observe that $\Bbb{Q}_p$ is not connected. This was a bit of a surprise to me as the completion of $\Bbb{Q}$ in the Archimedean metric is $\Bbb{R}$ which is connected.

This made me curious as to what are the connected components of $\Bbb{Q}_p$ and whether there is a characterization for the components in terms of the $p$-adic metric (or any characterization at all).

So finally my question is – What are the connected components of $\Bbb{Q}_p$ and what interesting information do they tell us? Further, how do we compare or contrast this situation with that of $\Bbb{R}$?

Best Answer

As a topological space $\Bbb{Q}_p$ has the following property. Given any two points $x,y\in\Bbb{Q}_p$ there exists open sets $U_1,U_2$ such that $U_1\cap U_2=\emptyset$, $U_1\cup U_2=\Bbb{Q}_p$, $x\in U_1$ and $y\in U_2$. Such a space is called totally disconnected.

To see that:

  1. Multiplication by any non-zero element is a homeomorphism, so we can assume that $x=1$ and $y\in\Bbb{Z}_p$ (switching the roles of $x$ and $y$ may be necessary here).
  2. If $m\ge0$ is the first $p$-adic digit where $y$ differs from $x$, then $y$ belongs to a coset $U_2$ of $p^m\Bbb{Z}_p$ that does not contain $x$.
  3. That $U_2$ is clopen, so we can use its complement as $U_1$.
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